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THE FOURTH DIMENSION

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FOURTH DIMENSION

C. HOWARD HINTON, M.A.

AUTHOR OF "SCIENTIFIC ROMANCES"

'• \ m:u i.ii a of thouoht," etc., etc-.

Loudon: SWAN SONNi;\s< ii kin a- CO., Ltd.

Nkw YOBM : JOHN LANE
1904

JUN 1 6 1933

PRINTED BY

HAZELL, WATSON AND VINI.Y, ID.,

LONDON AND AYLESBURY,

ENGLAND,

PREFACE

I have endeavoured to present the subject of the higher
dimensionality of space in a clear manner, devoid of
mathematical subtleties and technicalities. In order to
engage the interest of the reader, I have in the earlier
chapters dwelt on the perspective the hypothesis of a
fourth dimension opens, and have treated of the many
connections there are between this hypothesis and the
ordinary topics of our thoughts.

A lack of mathematical knowledge will prove of no
disadvantage to the reader, for I have used no mathe-
matical processes of reasoning. I have taken the view
that the space which we ordinarily think of, the space
of real things (which I would call permeable matter),
is different from the space treated of by mathematics.
Mathematics will tell us a great deal about space, just
as the atomic theory will tell us a great deal about the
chemical combinations of bodies. But after all, a theory
is not precisely equivalent to the subject with regard
to which it is held. There is an opening, therefore, from
the side of our ordinary space perceptions for a simple,
altogether rational, mechanical, and observational way

VI PREFACE

of treating this subject of higher space, and of this
opportunity I have availed myself.

The details introduced in the earlier chapters, especially
in Chapters VIII., IX., X., may perhaps be found
wearisome. They are of no essential importance in the
main line of argument, and if left till Chapters XI.
and XII. have been read, will be found to afford
interesting and obvious illustrations of the properties
discussed in the later chapters.

My thanks are due to the friends who have assisted
me in designing and preparing the modifications of
my previous models, and in no small degree to the
publisher of this volume, Mr. Sonnenschein, to whose
unique appreciation of the line of thought of this, as
of my former essays, their publication is owing. By
the provision of a coloured plate, in addition to the other
illustrations, he has added greatly to the convenience
of the reader.

C. Howard Hinton.

ERRATA.

Page 34, line 84, for B, read B'.

„ 61, „ 11, „ contract, „ construct.

„ 169, „ 20, ,, blue, „ purple.

„ 173, „ 3, „ eight, „ twelve.

„ 173, „ 5, „ 8 „ 12.

„ 173, „ 5, „ 20, „ 24.

CONTENTS

CHAP. PAGE

I. Four-Dimensional Space .... 1

II. The Analogy of a Plane World ... 6

III. The Significance of a Four-Dimensional

Existence . . . . . . .15

IV. The First Chapter in the History of Four

Space 23

V. The Second Chapter in the History of

Four Space. ...... 41

Lobatchewsky, Bolyai, and Gauss
Mttageometry

VI. The Higher World Gl

VII. The Evidences for a Fourth Dimension . 7G

VIII. The Use of Four Dimensions in Thought . 85

IX. Application to Kant's Theory of Experience 107

X. A Four-Dim bnsional Figure .... 122

XI. Nomenclature and Analogies . . L36

vii

Vlll CONTENTS

CHAP. PAGE

XII. Tiie Simplest Four-Dimensional Solid . .157

XIII. Remarks on the Figures . . . .178

XIV. A Recapitulation and Extension of the

Physical Argument 203

APPENDIX— The Models 231

THE FOURTH DIMENSION

CHAPTER I

FOUR DIMENSIONAL SPACE

There is nothing more indefinite, and at the same time
more real, than that which we indicate when we speak
of the " higher." In our social life we see it evidenced
in a greater complexity of relations. But this com-
plexity is not all. There is, at the same time, a contact
with, an apprehension of, something more fundamental,
more real.

With the greater development of man there comes
a consciousness of something more than all the forms
in which it shows itself. There is a readiness to give
up all the visible and tangible for the sake of those
principles and values of which the visible and tangible
are the representation. The physical life of civilised
man and of a mere savage are practically the same, but
the civilised man has discovered a depth in his existence,
which makes him feel that that which appears all to
the savage is a mere externality and appurtenage to ln's
true being.

.Now, this higher — how shall we apprehend it? It is
generally embraced by our religions faculties, by onr
idealising tendency. Bui the higher existence has two
H<lrs. It ha- a being as wvll as qualities. Ami in trying

Z THE FOURTH DIMENSION

to realise it through our emotions we are always taking the
subjective view. Our attention is always fixed on what we
feel, what we think. Is there any way of apprehending
the higher after the purely objective method of a natural
science ? I think that there is.

Plato, in a wonderful allegory, speaks of some men
living in such a condition that they were practically
reduced to be the denizens of a shadow world. They
wTere chained, and perceived but the shadows of them-
selves and all real objects projected on a wall, towards
which their faces were turned. All movements to them
were but movements on the surface, all shapes but the
shapes of outlines with no substantiality.

Plato uses this illustration to portray the relation
between true being and the illusions of the sense world.
He says that just as a man liberated from his chains
could learn and discover that the world was solid and
real, and could go back and tell his bound companions of
this greater higher reality, so the philosopher who has
been liberated, who has gone into the thought of the
ideal wTorld, into the world of ideas greater and more
real than the things of sense, can come and tell his fellow
men of that which is more true than the visible sun —
more noble than Athens, the visible state.

Now, I take Plato's suggestion ; but literally, not
metaphorically. He imagines a world which is lower
than this world, in that shadow figures and shadow
motions are its constituents ; and to it he contrasts the real
world. As the real world is to this shadow world, so is the
higher world to our world. I accept his analogy. As our
world in three dimensions is to a shadow or plane world,
so is the higher world to our three-dimensional world.
That is, the higher world is four-dimensional ; the higher
being is, so far as its existence is concerned apart from its
qualities, to be sought through the conception of an actual

FOUR-DIMENSIONAL SPACE 3

existence spatially higher than that which we realise with
our senses.

^ Here you will observe I necessarily leave out all that
gives its charm and interest to Plato's writings. All
those conceptions of the beautiful and good which live
immortally in his pages.

All that I keep from his great storehouse of wealth is
this one thing simply— a world spatially higher than this
world, a world which can only be approached through the
stocks and stones of it, a world which must be appre-
hended laboriously, patiently, through the material things
of it, the shapes, the movements, the figures of it.

We must learn to realise the shapes of objects in
this world of the higher man ; we must become familiar
with the movements that objects make in his world, so
that we can learn something about his daily experience,
his thoughts of material objects, his machinery.

The means for the prosecution of this enquiry are given
in the conception of space itself.

It often happens that that which we consider to be
unique and unrelated gives us, within itself, those relations
by means of which we are able to see it as related to
others, determining and determined by them.

Thus, on the earth is given that phenomenon of weight
by means of which Newton brought the earth into its
true relation to the sun and other planets. Our terrest rial
globe was determined in regard to other bodies of the
solar system by means of a relation which subsisted on
the earth itself.

And BO Bpace itself bears within it relations of which
We can determine it as related to otherspace. For within
Bpace are given the conceptions of point and line, line and
plane, which really involve the relation of space i<> a
higher space.

Where one segment of a straight line leaves off and

4 THE FOURTH DIMENSION

another begins is a point, and the straight line itself can
be generated by the motion of the point.

One portion of a plane is bounded from another by a
straight line, and the plane itself can be generated by
the straight line moving in a direction not contained
in itself.

Again, two portions of solid space are limited with
regard to each other by a plane ; and the plane, moving
in a direction not contained in itself, can generate solid
space.

Thus, going on, we may say that space is that which
limits two portions of higher space from each other, and
that our space will generate the higher space by moving
in a direction not contained in itself.

Another indication of the nature of four-dimensional
space can be gained by considering the problem of the
arrangement of objects.

If I have a number of swords of varying degrees of
brightness, I can represent them in respect of this quality
by points arranged along a straight line.

If I place a sword at a, fig. 1, and regard it as having
a certain brightness, then the other swords

can be arranged in a series along the

*ig- 1- imGj as at A) Bj Cj e|-c^ according to

their degrees of brightness.

If now I take account of another quality, say length,
they can be arranged in a plane. Starting from A, 13, c, I
can find points to represent different
E degrees of length along such lines as
af, BD, CE, drawn from A and B and C.
Points on these lines represent different
* lg- 2* degrees of length with the same degree of

brightness. Thus the whole plane is occupied by points
representing all conceivable varieties of brightness and
length.

FOUR-DIMENSIONAL SrACE O

Bringing in a third quality, say sharpness, I can draw,

as in fig. 3, any number of upright
lines. Let distances along these
upright lines represent degrees of
sharpness, thus the points F and G
will represent swords of certain
definite degrees of the three qualities
mentioned, and the whole of space will serve to represent
all conceivable degrees of these three qualities.

If now I bring in a fourth quality, such as weight, and
try to find a means of representing it as I did the other
three qualities, I find a difficulty. Every point in space is
taken up by some conceivable combination of the three
qualities already taken.

To represent four qualities in the same way as that in
which I have represented three, I should need another
dimension of space.

Thus we may indicate the nature of four-dimensional
space by saying that it is a kind of space which would
give positions representative of four qualities, as three-
dimensional space gives positions representative of three
qualities.

CHAPTER, II

THE ANALOGY OF A PLANE WORLD

At the risk of some prolixity I will go fully into the
experience of a hypothetical creature confined to motion
on a plane surface. By so doing I shall obtain an analogy
which will serve in our subsequent enquiries, because the
change in our conception, which we make in passing from
the shapes and motions in two dimensions to those in
three, affords a pattern by which we can pass on still
further to the conception of an existence in four-dimensional
space.

A piece of paper on a smooth table affords a ready
image of a two-dimensional existence. If we suppose the
being represented by the piece of paper to have no
knowledge of the thickness by which he projects above the
surface of the table, it is obvious that he can have no
knowledge of objects of a similar description, except by
the contact with their edges. His body and the objects
in his world have a thickness of which however, he has no
consciousness. Since the direction stretching up from
the table is unknown to him he will think of the objects
of his world as extending in two dimensions only. Figures
are to him completely bounded by their lines, just as solid
objects are to us by their surfaces. He cannot conceive
of approaching the centre of a circle, except by breaking
through the circumference, for the circumference encloses
the centre in the directions in which motion is possible to

THE ANALOGY OF A PLANE WORLD /

him. The plane surface over which he slips and with
which he is always in contact will be unknown to him ;
there are no differences by which he can recognise its
existence.

But for the purposes of our analogy this representation
is deficient.

A being as thus described has nothing about him to
push off from, the surface over which he slips affords no
means by which he can move in one direction rather than
another. Placed on a surface over wThich he slips freely,
he is in a condition analogous to that in which we should
be if we were suspended free in space. There is nothing
which he can push off from in any direction known to him.
Let us therefore modify our representation. Let us
suppose a vertical plane against which particles of thin
matter slip, never leaving the surface. Let these particles
possess an attractive force and cohere together into a disk ;
this disk will represent the globe of a plane being. He
must be conceived as existing on the rim.

Let 1 represent this vertical disk of flat matter and 2

the plane being on it, standing upon its

viin as we stand on the surface of our earth.

The direction of the attractive force of his

matter will give the creature a knowledge

of up and down, determining for him one

direction in his plane space. Also, since

1 "'-• 4- he can move along the surface of his earth,

he will have the sense of a direction parallel to its surface,

which we may call forwards and backwards.

He will have no sense of right and left — that is, of the
direction which we recognise as extending out from the
plane to our right and left.

The distinction of right and left is the one that we
must BUppose to be absent, in order to project ourselves
into the condition of a plane being.

8 THE FOURTH DIMENSION

Let the reader imagine himself, as he looks along the
plane, fig. 4, to become more and more identified with
the thin body on it, till he finally looks along parallel to
the surface of the plane earth, and up and down, losing
the sense of the direction which stretches right and left.
This direction will be an unknown dimension to him.

Our space conceptions are so intimately connected with
those which we derive from the existence of gravitation
that it is difficult to realise the condition of a plane being,
without picturing him as in material surroundings with
a definite direction of up and down. Hence the necessity
of our somewhat elaborate scheme of representation, which,
when its import has been grasped, can be dispensed with
for the simpler one of a thin object slipping over a
smooth surface, which lies in front of us.

It is obvious that we must suppose some means by
which the plane being is kept in contact with the surface
on which he slips. The simplest supposition to make is
that there is a transverse gravity, which keeps him to the
plane. This gravity must be thought of as different to
the attraction exercised by his matter, and as unperceived
by him.

At this stage of our enquiry I do not wish to enter
into the question of how a plane being could arrive at
a knowledge of the third dimension, but simply to in-
vestigate his plane consciousness.

It is obvious that the existence of a plane being must
be very limited. A straight line standing up from the
surface of his earth affords a bar to his progress. An
object like a wheel which rotates round an axis would
be unknown to him, for there is no conceivable way in
which he can get to the centre without going through
the circumference. He would have spinning disks, but
could not get to the centre of them. The plane being
can represent the motion from any one point of his space

THE ANALOGY OF A PLANE WORLD

Fijz. 5.

to any other, by means of two straight lines drawn at
right angles to each other.

Let ax and ay be two such axes. He can accomplish
the translation from A to B by going along ax to C, and
then from c along cb parallel to ay.

The same result can of course be obtained
by moving to d along ay and then parallel
to ax from D to B, or of course by any
diagonal movement compounded by these
axial movements.

By means of movements parallel to
these two axes he can proceed (except for
material obstacles) from any one point of his space to
any other.

If now we suppose a third line drawn
out from A at right angles to the plane
it is evident that no motion in either
of the two dimensions he knows will
carry him in the least degree in the
direction represented by A z.

The lines az and ax determine a
plane. If he could be taken off his plane, and trans-
ferred to the plane axz, he would be in a world exactly
like his own. From every line in his
world there goes off a space world exactly
like his own.

From every point in his world a line can
be drawn parallel to az in the direction
unknown to him. If we suppose the square
in fig. 7 to be a geometrical square from
every point of it, inside as well as on the
contour, a straight line can be drawn parallel
to az. The assemblage of these line- constitute a solid
figure, of which the Bquare in the plane is the base. If
we consider the Bquare to represent an object in the plain'

Fig. G.

Finr.

10 THE FOURTH DIMENSION

being's world then we must attribute to it a very small
thickness, for every real thing must possess all three
dimensions. This thickness he does not preceive, but
thinks of this real object as a geometrical square. He
thinks of it as possessing area only, and no degree of
solidity. The edges which project from the plane to a
very small extent he thinks of as having merely length
and no breadth — as being, in fact, geometrical lines.

With the first step in the apprehension of a third
dimension there would come to a plane being the con-
viction that he had previously formed a wrong conception
of the nature of his material objects. He had conceived
them as geometrical figures of two dimensions only.
If a third dimension exists, such figures are incapable
of real existence. Thus he would admit that all his real
objects had a certain, though very small thickness in the
unknown dimension, and that the conditions of his
existence demanded the supposition of an extended sheet
of matter, from contact with which in their motion his
objects never diverge.

Analogous conceptions must be formed by us on the
supposition of a four-dimensional existence. We must
suppose a direction in which we can never point extending
from every point of our space. We must draw a dis-
tinction between a geometrical cube and a cube of real
matter. The cube of real matter we must suppose to
have an extension in an unknown direction, real, but so
small as to be imperceptible by us. From every point
of a cube, interior as well as exterior, we must imagine
that it is possible to draw a line in the unknown direction.
The assemblage of these lines would constitute a higher
solid. The lines going off in the unknown direction from
the face of a cube would constitute a cube starting from
that face. Of this cube all that we should see in our
space would be the face.

THE ANALOGY OF A TLAXE WORLD 11

Again, just as the plane being can represent any
motion in his space by two axes, so we can represent any
motion in our three-dimensional space by means of three
axes. There is no point in our space to which we cannot
move by some combination of movements on the directions
marked out by these axes.

On the assumption of a fourth dimension we have
to suppose a fourth axis, which we will call AW. It must
be supposed to be at right angles to each and every
one of the three axes ax, ay, az. Just as the two axes,
AX, AZ, determine a plane which is similar to the original
plane on which we supposed the plane being to exist, but
which runs off from it, and only meets it in a line ; so in
our space if we take any three axes such as ax, ay, and
AW, they determine a space like our space world. This
space runs off from our space, and if we were transferred
to it we should find ourselves in a space exactly similar to
our own.

We must give up any attempt to picture this space in
its relation to ours, just as a plane being would have to
give up any attempt to picture a plane at right angles
to his plane.

Such a space and ours run in different directions from
the plane of ax and ay. They meet in this plane but
have nothing else in common, just as the plane space
of ax and AY and that of ax and az run in different
directions and have but the line ax in common.

Omitting all discussion of the manner on which a plane
being might be conceived to form a theory of a three-
dimensional existence, let us examine how, with the means
at his disposal, he could represent the properties of three-
dimensional objects.

There are two ways in which the plane being can think
of one of our solid bodies. II'- can think of the cube,
fig. 8, as composed of a number of sections parallel to

THE FOURTH DIMENSION

his plane, each lying

Fisr. 8.

in the third dimension a little
further off from his plane than
the preceding one. These sec-
tions he can represent as a
series of plane figures lying in
his plane, but in so representing
' them he destroys the coherence
*-^ of them in the higher figure.
The set of squares, a, b, c, d,
represents the section parallel
to the plane of the cube shown in figure, but they are
not in their proper relative positions.

The plane being can trace out a movement in the third
dimension by assuming discontinuous leaps from one
section to another. Thus, a motion along the edge of
the cube from left to right would be represented in the
set of sections in the plane as the succession of the
corners of the sections A, B, c, D. A point moving from
A through BCD in our space must be represented in the
plane as appearing in a, then in B, and so on, without
passing through the intervening plane space.

In these sections the plane being leaves out, of course,

the extension in the third dimension ; the distance between

any two sections is not represented. In order to realise

this distance the conception of motion can be employed.

Let fig. 9 represent a cube passing transverse to the

plane. It will appear to the plane being as a

square object, but the matter of which this

object is composed will be continually altering.

One material particle takes the place of another,

but it does not come from anywhere or go

anywhere in the space which the plane being

knows.

The analogous manner of representing a higher solid in
our case, is to conceive it as composed of a number of

Fisr. 9.

THE ANALOGY OF A PLANE WORLD

Fig:. 10.

sections, each lying a little further off in the unknown
direction than the preceding.

We can represent these sections as a number of solids.

Thus the cubes A, I?, C, D,
may be considered as
the sections at different
intervals in the unknown
dimension of a higher
cube. Arranged thus their coherence in the higher figure
is destroyed, they are mere representations.

A motion in the fourth dimension from A through B, C,
etc., would be continuous, but we can only represent it as
the occupation of the positions A, B, c, etc., in succession.
We can exhibit the results of the motion at different
stages, but no more.

In this representation we have left out the distance
between one section and another ; we have considered the
higher body merely as a series of sections, and so left out
its contents. The only way to exhibit its contents is to
call in the aid of the conception of motion.

If a higher cube passes transverse to our space, it will
appear as a cube isolated in space, the pait
that has not come into our space and the part
that has passed through will not be visible
The gradual passing through our space would
appear as the change of the matter of the cube
before us. One material particle in it is succeeded by
another, neither coming nor going in any direction we can
point to. In this manner, by the duration of the figure,
we can exhibit the higher dimensionality of it; a cube of
our matter, under the circumstances supposed, namely,
thai it has a motion transverse to our space, would instantly
disappear. A higher cube would last till it had passed
transverse to our space l»v its whole distance of extension
in the Fourl h dimension.

THK rolinil DIMKNSIuN

As the plane being can think of the cube as consisting
of sections, each like a figure he knows, extending away
from his plane, so we can think of a higher solid as com-
posed of sections, each like a solid which we know, but
extending away from our space.

Thus, taking a higher cube, we can look on it as
starting from a cube in our space and extending in the
unknown dimension.

Take the face A and conceive it to exist as simply a

Fig. 12.

face, a square with no thickness. From this face the
cube in our space extends by the occupation of space
which we can see.

But from this face there extends equally a cube in the
unknown dimension. We can think of the higher cube,
then, by taking the set of sections A, B, C, D, etc., and
considering that from each of them there runs a cube.
These cubes have nothing in common with each other,
and of each of them in its actual position all that we can
have in our space is an isolated square. It is obvious that
we can take our series of sections in any manner we
please. We can take them parallel, for instance, to any
one of the three isolated faces shown in the figure.
Corresponding to the three series of sections at right
angles to each other, which we can make of the cube
in space, wTe must conceive of the higher cube, as com-
posed of cubes starting from squares parallel to the faces
of the cube, and of these cubes all that exist in our space
are the isolated squares from which they start.

CHAPTER III

THE SIGNIFICANCE OF A FOUR-
DIMENSIONAL EXISTENCE

Having now obtained the conception of a four-dimensional
space, and having formed the analogy which, without
any further geometrical difficulties, enables us to enquire
into its properties, I will refer the reader, whose interest
is principally in the mechanical aspect, to Chapters VI.
and VI T. In the present chapter I will deal with the
general significance of the enquiry, and in the next
with the historical origin of the idea.

First, with regard to the question of whether there
is any evidence that we are really in four-dimensional
space, I will go back to the analogy of the plane world.

A being in a plane world could not have any ex-
perience of three-dimensional shapes, but he could have
an experience of three-dimensional movements.

We have seen that his matter must be supposed to
have an extension, though a very small one, in the third
dimension. And thus, in the small particles of his
matter, three-dimensional movements may well be <'<>n-
ceived to take place. Of these movements lie would only
perceive the resultants. Since all movements of an
observable size in the plane world are two-dimensional,
he would only perceive the resultant s in two dimensions
of t he small three-dimensional movements. Thus, there

would be phenomena Which lie could not explain by Ms

10 THE FOURTH DIMENSION

theory of mechanics — motions would take place which
he could not explain by his theory of motion. Hence,
to determine if we are in a four-dimensional world, we
must examine the phenomena of motion in our space.
If movements occur which are not explicable on the sup-
positions of our three-dimensional mechanics, we should
have an indication of a possible four-dimensional motion,
and if, moreover, it could be shown that such movements
would be a consequence of a four-dimensional motion in
the minute particles of bodies or of the ether, we should
have a strong presumption in favour of the reality of
the fourth dimension.

By proceeding in the direction of finer r.nd finer sub-
division, we come to forms of matter possessing properties
different to those of the larger masses. It is probable that
at some stage in this process we should come to a form
of matter of such minute subdivision that its particles
possess a freedom of movement in four dimensions. This
form of matter I speak of as four-dimensional ether, and
attribute to it properties approximating to those of a
perfect liquid.

Deferring the detailed discussion of this form of matter
to Chapter VI., we will how examine the means by which
a plane being would come to the conclusion that three-
dimensional movements existed in his world, and point
out the analogy by which we can conclude the existence
of four-dimensional movements in our world. Since the
dimensions of the matter in his world are small in the
third direction, the phenomena in which he would detect
the motion would be those of the small particles of
matter.

Suppose that there is a ring in his plane. We can
imagine currents flowing round the ring in either of two
opposite directions. These would produce unlike effects,
and give rise to two different fields of influence. If the

THE SIGNIFICANCE OF A FOUR- DIMENSIONAL EXISTENCE 17

ring with a current in it in one direction be taken up
and turned over, and put down again on the plane, it
would be identical with the ring having a current in the
opposite direction. An operation of this kind would be
impossible to the plane being. Hence he would have
in his space two irreconcilable objects, namely, the two
fields of influence due to the two rings with currents in
them in opposite directions. By irreconcilable objects
in the plane I mean objects which cannot be thought
of as transformed one into the other by any movement
in the plane.

Instead of currents flowing in the rings we can imagine
a different kind of current. Imagine a number of small
rings strung on the original ring. A current round these
secondary rings would give two varieties of effect, or two
different fields of influence, according to its direction.
These two varieties of current could be turned one into
the other by taking one of the rings up, turning it over,
and putting it down again in the plane. This operation
is impossible to the plane being, hence in this case also
there would be two irreconcilable fields in the plane.
Now, if the plane being found two such irreconcilable
fields and could prove that they could not be accounted
for by currents in the rings, he would have to admit the
existence of currents round the rings — that is, in rings
strung on the primary ring. Thus he would come to
admit the existence of a three-dimensional motion, for
such a deposition of currents is in three dimensions.

Now in our space there are two fields of different
properties, which can be produced by an electric current
flowing in a closed circuit or ring. These two fields can
he changed one into the other by reversing the currents, but
they cannot he changed one into the other by any turning
about of the rings in our space; for the disposition of the

field with regard to the ring itself is different when we

18 THE FOURTH DIMENSION

turn (lie ring, over and when we reverse the direction of
the current in the ring.

As hypotheses to explain the differences of these two
fields and their effects we can suppose the following kinds
of space motions : — First, a current along the conductor ;
second, a current round the conductor — that is, of rings of
currents strung on the conductor as an axis. Neither of
these suppositions accounts for facts of observation.

Hence we have to make the supposition of a four-
dimensional motion. We find that a four-dimensional
rotation of the nature explained in a subsequent chapter,
has the following characteristics : — First, it would give us
two fields of influence, the one of which could be turned
into the other by taking the circuit up into the fourth
dimension, turning it over, and putting it down in our
space again, precisely as the two kinds of fields in the
plane could be turned one into the other by a reversal of
the current in our space. Second, it involves a phenome-
non precisely identical with that most remarkable and
mysterious feature of an electric current, namely that it
is a field of action, the rim of which necessarily abuts on a
continuous boundary formed by a conductor. Hence, on
the assumption of a four-dimensional movement in the
region of the minute particles of matter, we should expect
to find a motion analogous to electricity.

Now, a phenomenon of such universal occurrence as
electricity cannot be due to matter and motion in any
very complex relation, but ought to be seen as a simple
and natural consequence of their properties. I infer that
the difficulty in its theory is due to the attempt to explain
a four-dimensional phenomenon by a three-dimensional
geometry.

In viewT of this piece of evidence we cannot disregard
that afforded by the existence of symmetry. In this
connection I will allude to the simple way of producing

THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 19

the images of insects, sometimes practised by children.
They put a few blots of ink in a straight line on a piece of
paper, fold the paper along the blots, and on opening it the
lifelike presentment of an insect is obtained. If we were
to find a multitude of these figures, we should conclude
that they had originated from a process of folding over ;
the chances against this kind of reduplication of parts
is too great to admit of the assumption that they had
been formed in any other way.

The production of the symmetrical forms of organised
beings, though not of course due to a turning over of
bodies of any appreciable size in four-dimensional space,
can well be imagined as due to a disposition in that
manner of the smallest living particles from which they
are built up. Thus, not only electricity, but life, and the
processes by which we think and feel, must be attributed
to that region of magnitude in which four-dimensional
movements take place.

I do not mean, however, that life can be explained as a
four-dimensional movement. It seems to me that the
whole bias of thought, which tends to explain the
phenomena of life and volition, as due to matter and
motion in some peculiar relation, is adopted rather in the
interests of the explicability of things than with any
regard to probability.

Of course, if we could show that life were a phenomenon
of motion, we should be able to explain a great deal that is
at present obscure. But there are two great difficulties in
flic way. It would be necessary to show that in a germ
capable of developing into a living being, there were
modifications of structure capable <>f determining in the
developed germ all the characteristics of its form, and nol
only this, lint of determining those of all the descendants
of Buch a foiin in ;m infinite Beries. Such a complexity of
mechanical relation-, undeniable though it be, cannot

20 THE FOURTH DIMENSION

surely be the best way of grouping the phenomena and
giving a practical account of them. And another difficulty
is this, that no amount of mechanical adaptation would
give that element of consciousness which we possess, and
which is shared in to a modified degree by the animal
world.

In those complex structures which men build up and
direct, such as a ship or a railway train (and which, if seen
by an observer of such a size that the men guiding them
were invisible, would seem to present some of the
phenomena of life) the appearance of animation is not
due to any diffusion of life in the material parts of the
structure, but to the presence of a living being.

The old hypothesis of a soul, a living organism within
the visible one, appears to me much more rational than the
attempt to explain life as a form of motion. And when we
consider the region of extreme minuteness characterised
by four-dimensional motion the difficulty of conceiving
such an organism alongside the bodily one disappears.
Lord Kelvin supposes that matter is formed from the
ether. We may very well suppose that the living
organisms directing the material ones are co-ordinate
with them, not composed of matter, but consisting of
etherial bodies, and as such capable of motion through
the ether, and able to originate material living bodies
throughout the mineral.

Hypotheses such as these find no immediate ground for
proof or disproof in the physical world. Let us, therefore,
turn to a different field, and, assuming that the human
soul is a four-dimensional being, capable in itself of four
dimensional movements, but in its experiences through
the senses limited to three dimensions, ask if the history
of thought, of these productivities which characterise man,
correspond to our assumption. Let us pass in review
those steps by which man, presumably a four-dimensional

THE SIGNIFICANCE OF A FOUR-DIMENSIONAL EXISTENCE 21

being, despite his bodily environment, has come to recog-
nise the fact of four-dimensional existence.

Deferring this enquiry to another chapter, I will here
recapitulate the argument in order to show that our
purpose is entirely practical and independent of any
philosophical or metaphysical considerations.

If two shots are fired at a target, and the second bullet
hits it at a different place to the first, we suppose that
there was some difference in the conditions under which
the second shot was fired from those affecting the first
shot. The force of the powder, the direction of aim, the
strength of the wind, or some condition must have been
different in the second case, if the course of the bullet was
not exactly the same as in the first case. Corresponding
to every difference in a result there must be some differ-
ence in the antecedent material conditions. By tracing
out this chain of relations we explain nature.

But there is also another mode of explanation which we
apply. If we ask what was the cause that a certain ship
was built, or that a certain structure was erected, we might
proceed to investigate the changes in the brain cells of
the men who designed the works. Every variation in one
ship or building from another ship or building is accom-
panied by a variation in the processes that go on in the
brain matter of the designers. But practically this would
be a very long task.

A more effective mode of explaining the production of
the ship or building would be to enquire into the motives,
plan- and aims of the men who constructed them. We
obtain a cumulative and consistent body of knowledge
much more easily and effectively in the latter way.

Sometimes we apply the one, sometimes the other
mode of explanation.

Bui it must be (iliMi'vul that the method of explana-
tion founded on aim, purpose, vol it ion, always presupposes

22 THE FOURTH DIMENSION

a mechanical system on which the volition and aim
works. The conception of man as willing and acting
from motives involves that of a number of uniform pro-
cesses of nature which he can modify, and of which he
can make application. In the mechanical conditions of
the three-dimensional world, the only volitional agency
which we can demonstrate is the human agency. But
when we consider the four-dimensional world the
conclusion remains perfectly open.

The method of explanation founded on purpose and aim
does not, surely, suddenly begin with man and end with
him. There is as much behind the exhibition of will and
motive which we see in man as there is behind the
phenomena of movement ; they are co-ordinate, neither
to be resolved into the other. And the commencement
of the investigation of that will and motive which lies
behind the will and motive manifested in the three-
dimensional mechanical field is in the conception of a
soul — a four-dimensional organism, which expresses its
higher physical being in the symmetry of the body, and
gives the aims and motives of human existence.

Our primary task is to form a systematic knowledge of
the phenomena of a four-dimensional world and find those
points in which this knowledge must be called in to
complete our mechanical explanation of the universe.
But a subsidiary contribution towards the verification of
the hypothesis may be made by passing in review the
history of human thought, and enquiring if it presents
such features as would be naturally expected on this
assumption.

CHAPTER IV

THE FIRST CHAPTER IN THE HISTORY
OF FOUR SPACE

Parmenides, and the Asiatic thinkers with whom he is
in close affinity, propound a theory of existence which
is in close accord with a conception of a possible relation
between a higher and a lower dimensional space. This
theory, prior and in marked contrast to the main stream
of thought, which we shall afterwards describe, forms a
closed circle by itself. It is one which in all ages has
had a strong attraction for pure intellect, and is the
natural mode of thought for those who refrain from
projecting their own volition into nature under the guise
of causality.

According to Parmenides of the school of Elea the all
is one, unmoving and unchanging. The permanent amid
the transient — that foothold for thought, that solid ground
for feeling on the discovery of which depends all our life —
is no phantom ; it is the image amidst deception of true
being, the eternal, the unmoved, the one. Thus says
Parmenides.

But how explain the shifting scene, these mutations
of things '

"Illusion," answers Parmenides. Distinguishing be-
tween truth and error, he tells of the true doctrine of the
one the false opinion of a changing world. He is no
memorable for the manner of bis advocacy than for

THE FOURTH DIMENSION

the cause he advocates. It is as if from his firm foothold
of being he could play with the thoughts under the
burden of which others laboured, for from him springs
that fluency of supposition and hypothesis which forms
the texture of Plato's dialectic.

Can the mind conceive a more delightful intellectual
picture than that of Parmenides, pointing to the one, the
true, the unchanging, and yet on the other hand ready to
discuss all manner of false opinion, forming a cosmogony
too, false u but mine own " after the fashion of the time?
In support of the true opinion he proceeded by the
negative way of showing the self-contradictions in the
ideas of change and motion. It is doubtful if his criticism,
save in minor points, has ever been successfully refuted.
To express his doctrine in the ponderous modern way we
must make the statement that motion is phenomenal,
not real.

Let us represent his doctrine.

Imagine a sheet of still water into which a slanting stick
is being lowered with a motion verti-
cally downwards. Let 1, 2, 3 (Fig. 13),
be three consecutive positions of the
stick. A, B, c, will be three consecutive
positions of the meeting of the stick,
with the surface of the water. As
the stick passes down, the meeting will
move from A on to B and C.

Suppose now all the water to be
removed except a film. At the meet-
ing of the film and the stick there
will be an interruption of the film.
If we suppose the film to have a pro-
perty, like that of a soap bubble, of closing up round any
penetrating object, then as the stick goes vertically
downwards the interruption in the film will move on.

Vis. 13.

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 25

If we pass a spiral through the film the intersection
will give a point moving in a circle shown by the dotted

lines in the figure. Suppose
now the spiral to be still and

ethe film to move vertically
upwards, the whole spiral will
be represented in the film of
the consecutive positions of the

v point of intersection. In the

\ film the permanent existence

j of the spiral is experienced as

I ^^^ a time series — the record of

^^^~^^ traversing the spiral is a point

F. moving in a circle. If now

we suppose a consciousness con-
nected with the film in such a way that the intersection of
the spiral with the film gives rise to a conscious experience,
we see that we shall have in the film a point moving in a
circle, conscious of its motion, knowing nothing of that
real spiral the record of the successive intersections of
which by the film is the motion of the point.

It is easy to imagine complicated structures of the
nature of the spiral, structures consisting of filaments,
and to suppose also that these structures are distinguish-
able from each other at every section. If we consider
the intersections of these filaments with the film as it
]>;i-s<>s to be the atoms constituting a filmar universe,
we shall have in the film a world of apparent motion;
we >liall have bodies corresponding to the filamentary
structure, and the positions of these structures with
regard to one another will give rise to bodies in the
film moving amongst one another. This mutual motion
i- apparent merely. The reality is of permanent structures
stationary, and all the relative motions accounted for by
one steady movement of the film as a whole

26 THE FOURTH DIMENSION

Thus we can imagine a plane world, in which all the
variety of motion is the phenomenon of structures con-
sisting of filamentary atoms traversed by a plane of
consciousness. Passing to four dimensions and our
space, we can conceive that all things and movements
in our world are the reading off of a permanent reality
by a space of consciousness. Each atom at every moment
is not what it was, but a new part of that endless line
which is itself. And all this system successively revealed
in the time which is but the succession of consciousness,
separate as it is in parts, in its entirety is one vast unity.
Representing Parmenides' doctrine thus, we gain a firmer
hold on it than if we merely let his words rest, grand and
massive, in our minds. And we have gained the means also
of representing phases of that Eastern thought to which
Parmenides was no stranger. Modifying his uncom-
promising doctrine, let us suppose, to go back to the plane
of consciousness and the structure of filamentary atoms,
that these structures are themselves moving — are acting,
living. Then, in the transverse motion of the film, there
would be two phenomena of motion, one due to the reading
off in the film of the permanent existences as they are in
themselves, and another phenomenon of motion due to
the modification of the record of the things themselves, by
their proper motion during the process of traversing them.

Thus a conscious being in the plane would have, as it
were, a two-fold experience. In the complete traversing
of the structure, the intersection of which with the film
gives his conscious all, the main and principal movements
and actions which he went through would be the record
of his higher self as it existed unmoved and unacting.
Slight modifications and deviations from these move-
ments and actions would represent the activity and self-
determination of the complete being, of his higher self.

It is admissible to suppose that the consciousness in

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 27

the plane has a share in that volition by which the
complete existence determines itself. Thus the motive
and will, the initiative and life, of the higher being, would
be represented in the case of the being in the film by an
initiative and a will capable, not of determining any great
things or important movements in his existence, but only
of small and relatively insignificant activities. In all the
main features of his life his experience would be repre-
sentative of one state of the higher being whose existence
determines his as the film passes on. But in his minute
and apparently unimportant actions he would share in
that will and determination by which the whole of the
being he really is acts and lives.

An alteration of the higher being would correspond to
a different life history for him. Let us now make the
supposition that film after film traverses these higher
structures, that the life of the real being is read off again
and again in successive waves of consciousness. There
would be a succession of lives in the different advancing
planes of consciousness, each differing from the preceding,
and differing in virtue of that will and activity which in
the preceding had not been devoted to the greater and
apparently most significant things in life, but the minute
and apparently unimportant. In all great things the
being of the film shares in the existence of his higher
self as it is at any one time. In the small things he
shares in that volition by which the higher being alters
and changes, acts and lives.

Thus we gain the conception of a life changing and
developing as a whole, a life in which our separation and
cessation and fugitiveness are merely apparent, but which
in its event- and course alters, changes, develops; and
the power of altering and changing this whole lies in the
will and power the limited being has of directing, guiding,
altering himself in the minute things of his existence.

28 THE FOURTH DIMENSION

Transferring our conceptions to those of an existence in

a higher dimensionality traversed by a space of con-
sciousness, we have an illustration of a thought which has
found frequent and varied expression. When, however,
we ask ourselves what degree of truth there lies in it, we
must admit that, as far as we can see, it is merely sym-
bolical. The true path in the investigation of a higher
dimensionality lies in another direction.

The significance of the Parmenidean doctrine lies in
this that here, as again and again, we find that those con-
ceptions which man introduces of himself, which he does
not derive from the mere record of his outward experience,
have a striking and significant correspondence to the
conception of a physical existence in a world of a higher
space. How close we come to Parmenides' thought by
this manner of representation it is impossible to say.
What I want to point out is the adequateness of the
illustration, not only to give a static model of his doctrine,
but one capable as it were, of a plastic modification into a
correspondence into kindred forms of thought. Either one
of two things must be true — that four-dimensional concep-
tions give a wonderful power of representing the thought
of the East, or that the thinkers of the East must have been
looking at and regarding four-dimensional existence.

Coming now to the main stream of thought we must
dwell in some detail on Pythagoras, not because of his
direct relation to the subject, but because of his relation
to investigators who came later.

Pythagoras invented the two-way counting. Let us
represent the single-way counting by the posits aa,
ab, ac, ad, using these pairs of letters instead of the
numbers 1, 2, 3, 4. I put an a in each case first for a
reason which will immediately appear.

We have a sequence and order. There is no con-
ception of distance necessarily involved. The difference

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 29

between the posits is one of order not of distance —

only when identified with a number of equal material

things in juxtaposition does the notion of distance arise.

Now, besides the simple series I can have, starting from

a<x, ba, ca, da, from ab, 66, cb, db, and so on, and forming

a scheme :

da db dc dd

ca cb co cd

ha lb be bd

aa ab ac ad

This complex or manifold gives a two-way order. I can
represent it by a set of points, if I am on my guard

• • • • against assuming any relation of distance.

• • • • Pythagoras studied this two-fold way of

• • • • counting in reference to material bodies, and

• • • • discovered that most remarkable property of

the combination of number and matter that
bears his name.
The Pythagorean property of an extended material
system can be exhibited in a manner which will be of
use to us afterwards, and which therefore I will employ
now instead of using the kind of figure which he himself
employed.

Consider a two-fold field of points arranged in regular
rows. Such a field will be presupposed in the following
argument.
• • o it is evident that in fig. 16 four

• t — t ■ f — j — | • of the points determine a square,

• 1 — 1 . n — ,, — „ . which square we may take as the

• • • . 1 — 1 — * • unit of measurement for areas.

But we can also measure areas
in another way.
Fig. 16 (1) Bhows foui- points determining a square.
But four squares also meet in a point, 6g. 16 (2).
Hence a point at the corner of a square belongs equally
to four Bquares.

30 THE FOURTH DIMENSION

Thus we may say that the point value of the square
shown is one point, for if we take the square in fig. 10(1)
it lias four points, but each of these belong equally to
four other squares. Hence one fourth of each of them
belongs to the square (1) in fig. 16. Thus the point
value of the square is one point.

The result of counting the points is the same as that
arrived at by reckoning the square units enclosed.

Hence, if we wish to measure the area of any square
we can take the number of points it encloses, count these
as one each, and take one-fourth of the number of points
at its corners.

Now draw a diagonal square as shown in fig. 17. It

contains one point and the four corners count for one

point more ; hence its point value is 2.

The value is the measure of its area — the

size of this square is two of the unit squares.

. j...i , . Looking now at the sides of this figure

we see that there is a unit square on each

£• 17* of them — the two squares contain no points,
but have four corner points each, which gives the point
value of each as one point.

Hence we see that the square on the diagonal is equal
to the squares on the two sides ; or as it is generally
expressed, the square on the hypothenuse is equal to
the sum of the squares on the sides.

Noticing this fact we can proceed to ask if it is always
true. Drawing the square shown in fig. 18, we can count
the number of its points. There are five
altogether. There are four points inside
the square on the diagonal, and hence, with
the four points at its corners the point
value is 5 — that is, the area is 5. Now
the squares on the sides are respectively
of the area 4 and 1. Hence in this case also the square

THE FIRST CHArTER IN THE HISTORY OF FOUR SrACE 31

on the diagonal is equal to the sum of the square on
the sides. This property of matter is one of the first
great discoveries of applied mathematics. We shall prove
afterwards that it is not a property of space. For the
present it is enough to remark that the positions in
which the points are arranged is entirely experimental.
It is by means of equal pieces of some material, or the
same piece of material moved from one place to another,
that the points are arranged.

Pythagoras next enquired what the relation must be
so that a square drawn slanting-wise should be equal to
one straight-wise. He found that a square whose side is
five can be placed either rectangularly along the lines
of points, or in a slanting position. And this square is
equivalent to two squares of sides 4 and 3.

Here he came upon a numerical relation embodied in
a property of matter. Numbers immanent in the objects
produced the equality so satisfactory for intellectual appre-
hension. And he found that numbers when immanent
in sound — when the strings of a musical instrument
were given certain definite proportions of Length — were
no less captivating to the ear than the equality of squares
was to the reason. What wonder then that he ascribed
an active power to number !

We must remember that, sharing like ourselves the
search for the permanent in changing phenomena, the
Greeks had not that conception of the permanent in
matter that we have. To them material things were not
permanent. In fire solid things would vanish ; absolutely
disappear. Rock and earth had a more stable existence,
but they too grew and decayed. The permanence of
matter, the conservation of energy, were unknown to
them. And that distinction which we (haw so readily

between the fleeting and permanent causes of sensation,
between a sound and a material object, for instance, had

32 THE FOURTH DIMENSION

not the same meaning to them which it has for us.
Let us but imagine for a moment that material things
are fleeting, disappearing, and we shall enter with a far
better appreciation into that search for the permanent
which, with the Greeks, as with us, is the primary
intellectual demand.

What is that which amid a thousand forms is ever the
same, which we can recognise under all its vicissitudes,
of which the diverse phenomena are the appearances ?

To think that this is number is not so very wide of
the mark. With an intellectual apprehension which far
outran the evidences for its application, the atomists
asserted that there were everlasting material particles,
which, by their union, produced all the varying forms and
states of bodies. But in view of the observed facts of
nature as then known, Aristotle, with perfect reason,
refused to accept this hypothesis.

He expressly states that there is a change of quality,
and that the change due to motion is only one of the
possible modes of change.

With no permanent material world about us, with
the fleeting, the unpermanent, all around we should, I
think, be ready to follow Pythagoras in his identification
of number with that principle wThich subsists amidst
all changes, which in multitudinous forms we apprehend
immanent in the changing and disappearing substance
of things.

And from the numerical idealism of Pythagoras there
is but a step to the more rich and full idealism of Plato.
That which is apprehended by the sense of touch we
put as primary and real, and the other senses we say
are merely concerned with appearances. But Plato took
them all as valid, as giving qualities of existence. That
the qualities were not permanent in the world as given
to the senses forced him to attribute to them a different

THE FIRST CHAPTER IN THE HISTORY OF FOUR SrACE 33

kind of permanence. He formed the conception of a
world of ideas, in which all that really is, all that affects
us and gives the rich and wonderful wealth of our
experience, is not fleeting and transitory, but eternal.
And of this real and eternal we see in the things about
us the fleeting and transient images.

And this world of ideas was no exclusive one, wherein
was no place for the innermost convictions of the soul and
its most authoritative assertions. Therein existed justice,
beauty — the one, the good, all that the soul demanded
to be. The world of ideas, Plato's wonderful creation
preserved for man, for his deliberate investigation and
their sure development, all that the rude incom-
prehensible changes of a harsh experience scatters and
destroys.

Plato believed in the reality of ideas. He meets us
fairly and squarely. Divide a line into two parts, he
says ; one to represent the real objects in the world, the
other to represent the transitory appearances, such as the
image in still water, the glitter of the sun on a bright
surface, the shadows on the clouds.

A B

Real things : Appearances:

e.g., the sun. e.g., the reflection of the sun.

Take another line and divide it into two parts, one
representing our ideas, the ordinary occupants of our
minds, such as whiteness, equality, and the other repre-
senting our true knowledge, which is of eternal principles,
sucli as beauty, goodness.

A1 ?>

Eternal principles, Appearances in tin- mind,

;i- beauty ;i> whiteness, < * j n .1 1 i t \-

Then as A is to B, so is A1 to B1.

That is, the soul can proceed, going away from real

34 THE FOURTH DIMENSION

tilings to a region of perfect certainty, where it beholds
what is, not the scattered reflections ; beholds the sun, not
the glitter on the sands ; true being, not chance opinion.

Now, this is to us, as it was to Aristotle, absolutely
inconceivable from a scientific point of view. We can
understand that a being is known in the fulness of his
relations ; it is in his relations to his circumstances that
a man's character is known ; it is in his acts under his
conditions that his character exists. We cannot grasp or
conceive any principle of individuation apart from the
fulness of the relations to the surroundings.

But suppose now that Plato is talking about the higher
man — the four-dimensional being that is limited in our
external experience to a three-dimensional world. Do not
his words begin to have a meaning ? Such a being
would have a consciousness of motion which is not as
the motion he can see with the eyes of the body. He,
in his own being, knows a reality to which the outward
matter of this too solid earth is flimsy superficiality. He
too knows a mode of being, the fulness of relations, in
which can only be represented in the limited world of
sense, as the painter unsubstantially portrays the depths
of woodland, plains, and air. Thinking of such a being
in man, was not Plato's line well divided ?

It is noteworthy that, if Plato omitted his doctrine of
the independent origin of ideas, he would present exactly
the four-dimensional argument; a real thing as we think
it is an idea. A plane being's idea of a square object is
the idea of an abstraction, namely, a geometrical square.
Similarly our idea of a solid thing is an abstraction, for in
our idea there is not the four-dimensional thickness which
is necessary, however slight, to give reality. The argu-
ment would then run, as a shadow is to a solid object, so
is the solid object to the reality. Thus A and B would
be identified.

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 35

In the allegory which I have already alluded to, Plato
in almost as many words shows forth the relation between
existence in a superficies and in solid space. And he
uses this relation to point to the conditions of a higher
being.

He imagines a number of men prisoners, chained so
that they look at the wall of a cavern in which they are
confined, with their backs to the road and the light.
Over the road pass men and women, figures and pro-
fessions, but of all this pageant all that the prisoners
behold is the shadow of it on the wall whereon they gaze.
Their own shadows and the shadows of the things in the
world are all that they see, and identifying themselves
with their shadows related as shadows to a world of
shadows, they live in a kind of dream.

Plato imagines one of their number to pass out from
amongst them into the real space world, and then return-
ing to tell them of their condition.

Here he presents most plainly the relation between
cxi.-tence in a plane world and existence in a three-
dimensional world. And he uses this illustration as a
type of the manner in which we are to proceed to a
higher Mate from the three-dimensional life we know.

It must have hung upon the weight of a shadow which
path he took! — whether the one we shall follow toward
the higher solid and the four-dimensional existence, or
the one which makes ideas the higher realities, and the
direct perception of them the contact with the truer
world.

Passing on to Aristotle, we will touch on the [joints
which most immediately concern our enquiry.

Jusl as a scientific man of the present day in
reviewing the -peculations of the ancient world would

treat them with a curiosity halt' amused hut wholly
respectful, asking of each and all wherein lay their

3G THE FOURTH DIMENSION

relation to fact, so Aristotle, in discussing the philosophy
of Greece as he found it, asks, above all other things :
11 Does this represent the world ? In this system is there
an adequate presentation of what is ? "

He finds them all defective, some for the very reasons
which we esteem them most highly, as when he criticises
the Atomic theory for its reduction of all change to motion.
But in the lofty march of his reason he never loses sight
of the whole ; and that wherein our views differ from his
lies not so much in a superiority of our point of view, as
in the fact which he himself enunciates — that it is im-
possible for one principle to be valid in all branches of
enquiry. The conceptions of one method of investigation
are not those of another ; and our divergence lies in our
exclusive attention to the conceptions useful in one way
of apprehending nature rather than in any possibility we
find in our theories of giving a view of the whole tran-
scending that of Aristotle.

He takes account of everything ; he does not separate
matter and the manifestation of matter ; he fires all
together in a conception of a vast world process in
which everything takes part — the motion of a grain of
dust, the unfolding of a leaf, the ordered motion of the
spheres in heaven — all are parts of one whole which
he will not separate into dead matter and adventitious
modifications.

And just as our theories, as representative of actuality,
fall before his unequalled grasp of fact, so the doctrine
of ideas fell. It is not an adequate account of exist-
ence, as Plato himself shows in his " Parmenides " ;
it only explains things by putting their doubles beside
them.

For his own part Aristotle invented a great marching
definition which, with a kind of power of its own, cleaves
its way through phenomena to limiting conceptions on

THE FIRST CHAPTER IN THE HISTORY OF FOUR SPACE 37

either hand, towards whose existence all experience
points.

In Aristotle's definition of matter and form as the
constituent of reality, as in Plato's mystical vision of the
kingdom of ideas, the existence of the higher dimension-
ality is implicitly involved.

Substance according to Aristotle is relative, not absolute.
In everything that is there is the matter of which it
is composed, the form which it exhibits ; but these are
indissolubly connected, and neither can be thought
without the other.

The blocks of stone out of which a house is built are the
material for the builder ; but, as regards the quarrymen,
they are the matter of the rocks with the form he has
imposed on them. Words are the final product of the
grammarian, but the mere matter of the orator or poet.
The atom is, with us, that out of which chemical substances
are built up, but looked at from another point of view is
the result of complex processes.

Nowhere do we find finality. The matter in one sphere
is the matter, plus form, of another sphere of thought.
Making an obvious application to geometry, plane figures
exist as the limitation of different portions of the plane
by one another. In the bounding lines the separated
matter of the plane shows its determination into form.
And as the plane is the matter relatively to determinations
in the plane, so the plane itself exists in virtue of tin'
determination of space. A plane is that wherein formless
space has form superimposed on it, and gives an actuality
of real relations. We cannot refuse to carry this process
of reasoning a Btep farther back, and say that space itself
i- thai which gives form to higher space. As a line is
the determination of a plane, and a plane of a solid, so
solid space itself is the determination of a higher space.

Afl a line by itself is inconceivable without that plane

38 Till-: FOURTH DIMENSION

which it separates, so the plane is inconceivable without
the solids which it limits on either hand. And so space
itself cannot be positively defined. It is the negation
of the possibility of movement in more than three
dimensions. The conception of space demands that of
a higher space. As a surface is thin and unsubstantial
without the substance of which it is the surface, so matter
itself is thin without the higher matter.

Just as Aristotle invented that algebraical method of
representing unknown quantities by mere symbols, not by
lines necessarily determinate in length as was the habit
of the Greek geometers, and so struck out the path
towards those objectifications of thought which, like
independent machines for reasoning, supply the mathe-
matician with his analytical weapons, so in the formulation
of the doctrine of matter and form, of potentiality and
actuality, of the relativity of substance, he produced
another kind of objectification of mind — a definition
which had a vital force and an activity of its own.

In none of his writings, as far as we kuow, did he carry it
to its legitimate conclusion on the side of matter, but in
the direction of the formal qualities he was led to his
limiting conception of that existence of pure form which
lies beyond all known determination of matter. The
unmoved mover of all things is Aristotle's highest
principle. Towards it, to partake of its perfection all
things move. The universe, according to Aristotle, is an
active process — he does not adopt the illogical conception
that it was once set in motion and has kept on ever since.
There is room for activity, will, self-determination, in
Aristotle's system, and for the contingent and accidental
as well. We do not follow him, because we are accus-
tomed to find in nature infinite series, and do not feel
obliged to pass on to a belief in the ultimate limits to
which they seem to point.

THE FIRST CHAPTER IN THE HISTORY OF FOUB SPACE 39

Bat apart from the pushing to the limit, as a relative
principle this doctrine of Aristotle's as to the relativity of
substance is irrefragible in its logic. He was the first to
show the necessity of that path of thought which when
followed leads to a belief in a four-dimensional space.

Antagonistic as he was to Plato in his conception
of the practical relation of reason to the world of
phenomena, yet in one point he coincided with him.
And in this he showed the candour of his intellect. He
was more anxious to lose nothing than to explain every-
thing. And that wherein so many have detected an
inconsistency, an inability to free himself from the school
of Plato, appears to us in connection with our enquiry
as an instance of the acuteness of his observation. For
beyond all knowledge given by the senses Aristotle held
that there is an active intelligence, a mind not the passive
recipient of impressions from without, but an active and
originative being, capable of grasping knowledge at first
hand. In the active soul Aristotle recognised something
in man not produced by his physical surroundings, some-
thing which creates, whose activity is a knowledge
underived from sense. This, he says, is the immortal and
undying being in man.

Thus we see that Aristotle was not far from the
recognition of the four-dimensional existence, both
without and within man, and the prut-ess of adequately
realising the higher dimensional figures to which we
shall come subsequently is a simple reduction to practice
of Id's livpoi hesia ol a ><>ul.

The next Btep in the unfolding of the drama of (lie
recognition of the soul as connected with our scientific
conception of the world, and, at the same time, (he
recognition of that higher of which a three-dimensional
world presents the superficial appearance, took place many
centuries Later. If we pass over the intervening time

40 THE FOURTH DIMENSION

without a word it is because the soul was occupied with
the assertion of itself in other ways than that of knowledge.
When it took up the task in earnest of knowing this
material world in which it found itself, and of directing
the course of inanimate nature, from that most objective
aim came, reflected back as from a mirror, its knowledge
of itself.

CHAPTER, V

THE SECOND CHAPTER IN THE HISTORY
OF FOUR SPACE

LOBATCHEWSKY, BOLYAI, AND GrAUSS

Before entering on a description of the work of
Lobatchewsky and Bolyai it will not be out of place
to give a brief account of them, the materials for which
are to be found in an article by Franz Schmidt in the
forty-second volume of the Mathematische Annalen,
and in EngeFs edition of Lobatchewsky.

Lobatchewsky was a man of the most complete and
wonderful talents. As a youth he was full of vivacity,
carrying his exuberance so far as to fall into serious
trouble for hazing a professor, and other freaks. Saved
by the good offices of the mathematician Bartels, who
appreciated his ability, he managed to restrain himself
within the bounds of prudence. Appointed professor at
his own University, Kasan, he entered on his duties under
the regime of a pietistic reactionary, who surrounded
himself with sycophants and hypocrites. Esteeming
probably the interests of his pupils as higher than any
attempt at a vain resistance, he made himself the tyrant's
light-hand man, doing an incredible amount of teaching
and performing the most varied official duties. Amidst
all his activities lie found time to make important con-
tributions to science. His theory of parallels is most

42 I'll 10 FOURTH DIMENSION.

closely connected with Lis name, but a study of bis
writings shows that he was a man capable of carrying
on mathematics in its main lines of advance, and of a
judgment equal to discerning what these lines were.
Appointed rector of his University, he died at an
advanced age, surrounded by friends, honoured, with the
results of his beneficent activity all around him. To him
no subject came amiss, from the foundations of geometry
to the improvement of the stoves by which the peasants
warmed their houses.

He was born in 1793. His scientific work was
unnoticed till, in 1867, Houel, the French mathematician,
drew attention to its importance.

Johann Bolyai de Bolyai was born in Klausenburg,
a town in Transylvania, December loth, 1802.

His father, Wolfgang Bolyai, a professor in the
Reformed College of Maros Vasarhely, retained the ardour
in mathematical studies which had made him a chosen
companion of Gauss in their early student days at
Gottingen.

He found an eager pupil in Johann. He relates that
the boy sprang before him like a devil. As soon as he
had enunciated a problem the child would give the
solution and command him to go on further. As a
thirteen-year-old boy his father sometimes sent him to fill
his place when incapacitated from taking his classes.
The pupils listened to him with more attention than to
his father for they found him clearer to understand.

In a letter to Gauss Wolfgang Bolyai writes : —

" My boy is strongly built. He has learned to recognise
many constellations, and the ordinary figures of geometry.
He makes apt applications of his notions, drawing for
instance the positions of the stars with their constellations.
Last winter in the country, seeing Jupiter he asked :
1 How is it that we can see him from here as well as from

TfiE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 43

the town ? He must be far off.' And as to three
different places to which he had been he asked me to tell
him about them in one word. I did not know what he
meant, and then he asked me if one was in a line with
the other and all in a row, or if they were in a triangle.

11 He enjoys cutting paper figures with a pair of scissors,
and without my ever having told him about triangles
remarked that a right-angled triangle which he had cut
out was half of an oblong. I exercise his body with care,
he can dig well in the earth with his little hands. The
blossom can fall and no fruit left. When he is fifteen
I want to send him to you to be your pupil."

In Johann's autobiography he sajTs : —

"My father called my attention to the imperfections
and gaps in the theory of parallels. He told me he had
gained more satisfactory results than his predecessors,
but had obtained no perfect and satisfying conclusion.
None of his assumptions had the necessary degree of
geometrical certainty, although they sufficed to prove the
eleventh axiom and appeared acceptable on first sight.

" He begged of me, anxious not without a reason, to
hold myself aloof and to shun all investigation on this
subject, if I did not wish to live all my life in vain."

Johann, in the failure of his father to obtain any
response from Gauss, in answer to a letter in which he
asked the great mathematician to make of his son " an
apostle of truth in afar land," entered the Engineering
School at Vienna, lie writes from Temesvar, where he
was appointed sub-lieutenant September, 1823: —

"Temesvar, November 3rd, L823.
" Dear Good Father,

"I have so overwhelmingly much to write
about my discovery thai I know no other way of checking
myself than taking a quarter of a Bbeel only to write on.
1 want an answer to my four-sheet Letter.

44 THE FOURTH DIMENSION

"I am unbroken in my determination to publish a
work on Parallels, as soon as I have put my material in
order and have the means.

" At present I have not made any discovery, but
the way I have followed almost certainly promises me
the attainment of my object if any possibility of it
exists.

" I have not got my object yet, but I have produced
such stupendous things that I was overwhelmed myself,
and it would be an eternal shame if they were lost.
When you see them you will find that it is so. Now
I can only say that I have made a new world out of
nothing. Everything that I have sent you before is a
house of cards in comparison with a tower. I am con-
vinced that it will be no less to my honour than if I had
already discovered it."

The discovery of which Johann here speaks was
published as an appendix to Wolfgang Bolyai's Tentamen.
Sending the book to Gauss, Wolfgang writes, after an
interruption of eighteen years in his correspondence: —
" My son is first lieutenant of Engineers and will soon
be captain. He is a fine youth, a good violin player,
a skilful fencer, and brave, but has had many duels, and
is wild even for a soldier. Yet he is distinguished — light
in darkness and darkness in light. He is an impassioned
mathematician with extraordinary capacities. . . . He
will think more of your judgment on his work than that
of all Europe."

Wolfgang received no answer from Gauss to this letter,
but sending a second copy of the book received the
following reply : — :

" You have rejoiced me, my un forgotten friend, by your
letters. I delayed answering the first because I wanted
to wait for the arrival of the promised little book.
"Now something about your son's work.

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 45

" If I begin with saying that ' I ought not to praise it,'
you will be staggered for a moment. But I cannot say
anything else. To praise it is to praise myself, for the
path your son has broken in upon and the results to which
he has been led are almost exactly the same as my own
reflections, some of which date from thirty to thirty-five
years ago.

" In fact I am astonished to the uttermost. My inten-
tion was to let nothing be known in my lifetime about
my own work, of which, for the rest, but little is com-
mitted to writing. Most people have but little perception
of the problem, and I have found very few who took any
interest in the views I expressed to them. To be able to
do that one must first of all have had a real live feeling
of what is wanting, and as to that most men are com-
pletely in the dark.

" Still it was my intention to commit everything to
writing in the course of time, so that at least it should
not perish with me.

" I am deeply surprised that this task can be spared
me, and I am most of all pleased in this that it is the son
of my old friend who has in so remarkable a manner
preceded me."

The impression which we receive from Gauss's in-
explicable silence towards his old friend is swept away
by this letter. Hence we breathe the clear air of the
mountain tops. Gauss would not have failed to perceive
the vast significance of his thoughts, sure to be all the
greater in their effect on future ages from the want of
comprehension of llie present. Yet there is not a word
or a sign in his writing to claim the thought for himself.
He published do single line on the subject. P>y the
measure of what lie thus silently relinquishes, by such a
measure of a world-transforming t bought , we can appre-
ciate his great ne

4G THE FOURTH DIMENSION

It is a long step from Gauss's serenity to the disturbed
and passionate life of Johann Bolyai — he and Galois,

the two most interesting figures in the history of mathe-
matics. For Bolyai, the wild soldier, the duellist, fell
at odds with the world. It is related of him that he was
challenged by thirteen officers of his garrison, a thing not
unlikely to happen considering how differently he thought
from every one else. He fought them all in succession —
making it his only condition that he should be allowed
to play on his violin for an interval between meeting each
opponent. He disarmed or wounded all his antagonists.
It can be easily imagined that a temperament such as
his was one not congenial to his military superiors. He
was retired in 1833.

His epoch-making discovery awoke no attention. He
seems to have conceived the idea that his father had
betrayed him in some inexplicable way by his communi-
cations with Gauss, and he challenged the excellent
Wolfgang to a duel. He passed his life in poverty,
many a time, says his biographer, seeking to snatch
himself from dissipation and apply himself again to
mathematics. But his efforts had no result. He died
January 27th, 1860, fallen out with the world and with
himself.

Metageometry

The theories which are generally connected with the
names of Lobatchewsky and Bolyai bear a singular and
curious relation to the subject of higher space.

In order to show what this relation is, I must ask the
reader to be at the pains to count carefully the sets of
points by which I shall estimate the volumes of certain
figures.

THE SECOND CHAPTER IN THE HISTOltY OF FOUE SFACE 47

No mathematical processes beyond this simple one of
• • . . counting will be necessary.
. . • . Let us suppose we have before us in
. . . • fig. 19 a plane covered with points at regular
. . . . intervals, so placed that every four deter-
mine a square.

Now it is evident that as four points

Fig. 19.

determine a square, so four squares meet in a point.

•ffl

• ■ • i>

Thus, considering a point inside a square as
belonging to it, we may say that a point on
the corner of a square belongs to it and to
four others equally : belongs a quarter of it
to each square.
Thus the square acde (fig. 21) contains one point, and

Fig. 20.

Fig. 21.

• Ol • • •

v >

i — i

•

* A

B i

•

>— i

i •

•

Fie. 22.

lias four points at the four corners. Since one-fourth of
each of these four belongs to the square, the four together
count as one point, and the point value of the square is
two points — the one inside and the four at the corner
malm two points belonging to it exclusively.

Now the area of this square is two unit squares, as can
•i-ii by diawing two diagonals in fig. 22.

We also notice that the square in question is equal to
the sum of the squares on tin' Bides ab, v>(\ of the right-
angled triangle abc. Tims wo recognise the proposition
thai the square on the hypothenuse is equal to the sum
of the squares on the two sides of a right-angled triangle.

Now suppose we set ourselves the question of deter-
mining the whereabouts in the ordered system of points,

THE FOURTH DIMENSION

the end of a line would come when it turned about a
point keeping one extremity fixed at the point.

We can solve this problem in a particular case. If we
can find a square lying slantwise amongst the dots which is
equal to one which goes regularly, we shall know that the
two sides are equal, and that the slanting side is equal to the
straight-way side. Thus the volume and shape of a figure
remaining unchanged will be the test of its having rotated
about the point, so that we can say that its side in its first

turn into its side in the second position.

square can be found in the one whose side

length.

position would

Now, such a

is five units in

Fig. 23.
In fig. 23, in the square on ab,

lere are-

9 points interior .....

4 at the corners

4 sides with 3 on each side, considered as

I?; on each side, because belonging

equally to two squares

The total is 1C. There are 9 points in the square
on bc.

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 49

In the square on AC there are —

2 i points inside ..... 24
1 at the corners ..... 1

or 25 altogether.

Hence we see again that the square on the hypothenuse
is equal to the squares on the sides.

Now take the square AFHG, which is larger than the
square on ab. It contains 25 points.

16 inside 16

16 on the sides, counting as . . .8
4 on the corners ..... 1

making 25 altogether.

If two squares are equal we conclude the sides are
equal. Hence, the line AF turning round A would
move so that it would after a certain turning coincide
with AC.

This is preliminary, but it involves all the mathematical
difficulties that will present themselves.

There are two alterations of a body by which its volume
is not changed.

One is the one we have just considered, rotation, the
other is what is called shear.

Consider a book, or heap of loose pages. They can be

slid so that each one slips
over the preceding one,
a \> and the whole assumes

the shape b in fig. 24.
Tin's deformation is not shear alone, but shear accom-
panied by rotation.

Shear can be considered as produced in another way.
Take (lie square abcd (fig. 25), and suppose <lial it
i- pulled <>iit from along one of its diagonals both ways,
and proportionately compressed along the other diagonal.
It will assume the Bhape in fig. 2(1.

THE FOURTH DIMENSION

This compression and expansion along two lines at right
angles is what is called shear; it is equivalent to the
sliding illustrated above, combined with a turning round.

Fig. 26.

In pure shear a body is compressed and extended in
two directions at right angles to each other, so that its
volume remains unchanged.

Now we know that our material bodies resist shear —
shear does violence to the internal arrangement of their
particles, but they turn as wholes without such internal
resistance.

But there is an exception. In a liquid shear and
rotation take place equally easily, there is no more
resistance against a shear than there is against a
rotation.

Now, suppose all bodies were to be reduced to the liquid
state, in which they yield to shear and to rotation equally
easily, and then were to be reconstructed as solids, but in
such a way that shear and rotation had interchanged
places.

That is to say, let us suppose that when they had
become solids again they would shear without offering
any internal resistance, but a rotation would do violence
to their internal arrangement.

That is, we should have a world in which shear would
have taken the place of rotation.

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 51

A shear does not alter the volume of a body : thus an
inhabitant living in such a world would look on a body
sheared as we look on a body rotated. He would say
that it was of the same shape, but had turned a bit
round.

Let us imagine a Pythagoras in this world going to
work to investigate, as is his wont.

Fig. 27 represents a square un sheared. Fig. 28

•

• •

>/D-

. . E/^/- •

• • / • j

z\ '

•A

far

B '

• i

•

t • •

Fig. 27.

Fig. 23.

represents a square sheared. It is not the figure into
which the square in fig. 27 would turn, but the result of
shear on some square not drawn. It is a simple slanting
placed figure, taken now as we took a simple slanting
] (laced xjuare before. Now, since bodies in this world of
shear otter no internal resistance to shearing, and keep
their volume when sheared, an inhabitant accustomed to
them would not consider that they altered their shape
under shear. He would call acde as much a square as
the square in fig. 27. AVe will call such figures shear
squares. Counting the dots in ACDE, we find —

2 inside = 2

4 at corners = 1

Or a total of 3.

.\<.\v, the Bqiiare On the side All has 4 points, that on 150

ha- 1 point. Hero the .-hear Bquare on the hypothenu.se
ha- not 5 points but 3; i( is not the sum of the squares on
t he Bides, bul t he different

THE FOURTH DIMENSION

This relation always holds. Look at
fig. 29.

Shear square on hypothenuse —

7 internal
4 at corners

Fig. 29.

Square on one side — which the reader can draw for

himself —

• <

• F

• • • *^^

• • / • • i

• » • •

.♦ • • •

Fig. 29 Z

'is.

4 internal .
8 on sides .
4 at corners

and the square on the other
side is 1. Hence in this
case again the difference is
equal to the shear square on
the hypothenuse, 9—1 = 8.
Thus in a world of shear
the square on the hypothen-
use would be equal to the
difference of the squares on
the sides of a right-angled
triangle.

In fig. 29 bis another shear square is drawn on which
the above relation can be tested.

What now would be the position a line on turning by
shear would take up ?

We must settle this in the same way as previously with
our turning.

Since a body sheared remains the same, we must find two
equal bodies, one in the straight way, one in the slanting
way, which have the same volume. Then the side of one
will by turning become the side of the other, for the two
figures are each what the other becomes by a shear turning.

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 53

We can solve the problem in a particular case —

In the figure
(fig. 30) there are —

f • •

• •

ACDE

15 inside .
4 at corners

. 15

a total of 16.

Now in the square abgf,
there are 16 —

1) inside
12 on sides .
4 at corners

. 9

. 6
. 1

Hence the square on AB
would, by the shear turn-
ing, become the shear square
Fig. 30. 5 ^

° ACDE.

And hence the inhabitant of this world would say that
the line ab turned into the line AC. These two lines
would be to him two lines of equal length, one turned
a little way round from the other.

That is, putting shear in place of rotation, we get a
different kind of figure, as the result of the shear rotation,
from what we got with our ordinary rotation. And as a
consequence we get a position for the end of a line of
invariable length when it turns by the shear rotation,
different from the position which it would assume on
turning by our rotation.

A real material rod in the shear world would, on turning
about A, pass from the position ab to the position AC.
We say that its length alters when it becomes AC, but this
transformation of ab would seem to an inhabitant of the
Bheai world like a turning of AB without altering in
Length.

If now we Buppose a communication of ideas thai takes
place between one of ourselves and an inhabitant of the

T)4 THE FOURTH DIMENSION

shear world, there would evidently be a difference between
his views of distance and ours.

We should say that his line ah increased in length in
turning to AC. He would say thai our line ae (tig. 23)
decreased in length in turning to AC. lie would think
that what we called an equal line was in reality a shorter
one.

We should say that a rod turning round would have its
extremities in the positions we call at equal distances.
So would he — but the positions would be different. He
could, like us, appeal to the properties of matter. His
rod to him alters as little as ours does to us.

Now, is there any standard to which we could appeal, to
say which of the two is right in this argument ? There
is no standard.

We should say that, with a change of position, the
configuration and shape of his objects altered. He would
say that the configuration and shape of our objects altered
in what we called merely a change of position. Hence
distance independent of position is inconceivable, or
practically distance is solely a property of matter.

There is no principle to which either party in this
controversy could appeal. There is nothing to connect
the definition of distance with our ideas rather than with
his, except the behaviour of an actual piece of matter.

For the study of the processes which go on in our world
the definition of distance given by taking the sum of the
squares is of paramount importance to us. But as a ques-
tion of pure space without making any unnecessary
assumptions the shear world is just as possible and just as
interesting as our world.

It was the geometry of such conceivable worlds that
Lobatchewsky and Bolyai studied.

This kind of geometry has evidently nothing to do
directly with four-dimensional space,

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 55

But a connection arises in this way. It is evident that,
instead of taking a simple shear as I have done, and
defining it as that change of the arrangement of the
particles of a solid which they will undergo without
offering any resistance due to their mutual action, I
might take a complex motion, composed of a shear and
a rotation together, or some other kind of deformation.

Let us suppose such an alteration picked out and
defined as the one which means simple rotation, then the
type, according to which all bodies will alter by this
rotation, is fixed.

Looking at the movements of this kind, we should say
that the objects were altering their shape as well as
rotating. But to the inhabitants of that world they
would seem to be unaltered, and our figures in their
motions would seem to them to alter.

In such a world the features of geometry are different.
We have seen one such difference in the case of our illus-
tration of the world of shear, where the square on the
hypothenuse was equal to the difference, not the sum, of
the squares on the sides.

In our illustration we have the same laws of parallel
lines as in our ordinary rotation world, but in general the
laws of parallel lines are different.

In one of these worlds of a different constitution of
matter through one point there can be two parallels to
a given line, in another of them there can be none, that
is, although a line be drawn parallel to another it will
meet it after a time.

Now it was precisely in this respect of parallels that
Lobatchewsky and Bolyai discovered these different
worlds. They did not think of them as worlds of matter,
hut they discovered that -pace (lid not necessarily mean
that our law of parallels is true. They made the

distinction between laws of Bpace and law- of matter,

56 THE FOUBTH DIMENSION

although that is not the form in which they slated their
results.

The way in which they were led to these results was the
following. Euclid had stated the existence of parallel lines
as a postulate — putting frankly this unproved proposition
— that one line and only one parallel to a given straight
line can be drawn, as a demand, as something that must
be assumed. The words of his ninth postulate are these :
" If a straight line meeting two other straight lines
makes the interior angles on the same side of it ecmal
to two right angles, the two straight lines will never
meet."

The mathematicians of later ages did not like this bald
assumption, and not being able to prove the proposition
they called it an axiom — the eleventh axiom.

Many attempts were made to prove the axiom ; no one
doubted of its truth, but no means could be found to
demonstrate it. At last an Italian, Sacchieri, unable to
find a proof, said : "Let us suppose it not true." He deduced
the results of there being possibly two parallels to one
given line through a given point, but feeling the waters
too deep for the human reason, he devoted the latter half
of his book to disproving what he had assumed in the first
part.

Then Bolyai and Lobatchewsky with firm step entered
on the forbidden path. There can be no greater evidence
of the indomitable nature of the human spirit, or of its
manifest destiny to conquer all those limitations which
bind it down within the sphere of sense than this grand
assertion of Bolyai and Lobatchewsky.

~ ?v Take a line ah and a point C. We

say and see and know that through c

r d can only be drawn one line parallel

Fig. 31. to AB.

But Bolyai said ; " I will draw two." Let CD be parallel

THE SECOND CHArTER IN THE HISTORY OF FOUR SPACE 57

to AB, that is, not meet ab however far produced, and let

lines beyond CD also not meet

ab; let there be a certain

region between CD and CF,

in which no line drawn meets

lg" ,J,2< ab. CE and CD produced

backwards through c will give a similar region on the

other side of c.

Nothing so triumphantly, one may almost say so
insolently, ignoring of sense had ever been written before.
Men had struggled against the limitations of the body,
fought them, despised them, conquered them. But no
one had ever thought simply as if the body, the bodily
eyes, the organs of vision, all this vast experience of space,
had never existed. The age-long contest of the soul with
the body, the struggle for mastery, had come to a cul-
mination. Bolyai and Lobatchewsky simply thought as
if the body was not. The struggle for dominion, the strife
and combat of the soul were over; they had mastered,
and the Hungarian drew his line.

Can we point out any connection, as in the case of
Parmenides, between these speculations and higher
space? Can we suppose it was any inner perception by
the soul of a motion not known to the senses, which re-
sulted in this theory so free from the bonds of sense ? No
such supposition appears to be possible.

Practically, however, metageometry had a great in-
fluence in bringing the higher space to the front as a
working hypothesis. This can be traced to the tendency
the Tiiind has t<» move in the direction of least resistance.
The results of the new geometry could not be neglected,
1 Ik- problem of parallels had occupied a place too prominent
in the development of mathematical thought for its final
solution i.i be neglected. But this utter independence of
all mechanical considerations, this perfecl cutting Loose

f)8 THE FOURTH DIMENSION

from the familiar intuitions, was so difficult that almost
any other hypothesis was more easy of acceptance, and
when Beltrami showed that the geometry of Lobatche\\>kv
and Bolyai was the geometry of shortest lines drawn on
certain curved surfaces, the ordinary definitions of measure-
ment being retained, attention was drawn to the theory of
a higher space. An illustration of Beltrami's theory is
furnished by the simple consideration of hypothetical
beings living on a spherical surface.

Let A BCD be the equator of a globe, and AP, bp,
meridian lines drawn to the pole, P.
The lines ab, ap, bp would seem to be
perfectly straight to a person moving
on the surface of the sphere, and
unconscious of its curvature. Now
AP and bp both make right angles
P' with ab. Hence they satisfy the

Fig. 33. definition of parallels. Yet they

meet in P. Hence a being living on a spherical surface,
and unconscious of its curvature, wTould find that parallel
lines would meet. He would also find that the angles
in a triangle were greater than two right angles. In
the triangle pab, for instance, the angles at A and B
are right angles, so the three angles of the triangle
PAB are greater than two right angles.

Now in one of the systems of metageometry (for after
Lobatchewsky had shown the way it was found that other
systems were possible besides his) the angles of a triangle
are greater than two right angles.

Thus a being on a sphere would form conclusions about
his space which are the same as he would form if he lived
on a plane, the matter in which had such properties as
are presupposed by one of these systems of geometry.
Beltrami also discovered a certain surface on which there
could be drawn more than one " straight " line through a

THE SECOND CHAPTER IN THE HISTORY OF FOUR SPACE 59

point which would not meet another given line. I use
the word straight as equivalent to the line having the
property of giving the shortest path between any two
points on it. Hence, without giving up the ordinary
methods of measurement, it was possible to find conditions
in which a plane being would necessarily have an ex-
perience corresponding to Lobatchewsky's geometry.
And by the consideration of a higher space, and a solid
curved in such a higher space, it was possible to account
for a similar experience in a space of three dimensions.

Now, it is far more easy to conceive of a higher dimen-
sionality to space than to imagine that a rod in rotating
does not move so that its end describes a circle. Hence,
a logical conception having been found harder than that
of a four dimensional space, thought turned to the latter
as a simple explanation of the possibilities to which
Lobatchewsky had awakened it. Thinkers became accus-
tomed to deal with the geometry of higher space — it was
Kant, says Veronese, who first used the expression of
u different spaces " — and with familiarity the inevitable-
ness of the conception made itself felt.

From this point it is but a small step to adapt the
ordinary mechanical conceptions to a higher spatial
existence, and then the recognition of its objective
existence could be delayed no longer. Here, too, as in so
many cases, it turns out that the order and connection of
our ideas is the order and connection of things.

What is the significance of Lobatchewsky's and Bolyai's
work ?

It must be recognised as something totally different
from the conception of a higher space; it is applicable to
Bpaces of any number of dimensions. By immersing (he
concept ion of Hi-tance in matter to which it properly
belongs, it promises to be of the greatest aid in analysis
for the effective distance of any two particles is the

CO THE FOURTH DIMENSION

product of complex material conditions and cannot be
measured by Lard and fast rules. Its ultimate signi-
ficance is altogether unknown. It is a cutting loose
from the bonds of sense, not coincident with the recognition
of a higher dimensionality, but indirectly contributory
thereto.

Thus, finally, we have come to accept what Plato held
in the hollow of his hand ; what Aristotle's doctrine of
the relativity of substance implies. The vast universe, too,
has its higher, and in recognising it we find that the
directing being within us no longer stands inevitably
outside our systematic knowledge.

CHAPTER VI

THE HIGHER WORLD

It is indeed strange, the manner in which we must begin
to think about the higher world.

Those simplest objects analogous to those which are
about us on every side in our daily experience such as a
door, a table, a wheel are remote and un cognisable in the
world of four dimensions, while the abstract ideas of
rotation, stress and strain, elasticity into which analysis
resolves the familiar elements of our daily experience are
transferable and applicable with no difficulty whatever.
Thus we are in the unwonted position of being obliged
to contrast the daily and habitual experience of a four-
dimensional being, from a knowledge of the abstract
theories of the space, the matter, the motion of it ;
instead of, as in our case, passing to the abstract theories
from the richness of sensible things.

What would a wheel be in four dimensions? What
the shafting for the transmission of power which a
four-dimensional being would use.

The four-dimensional wheel, and the four-dimensional
shafting are what will occupy us for these few pages. And
it is no futile or insignificant enquiry. For in the attempt
to penetrate into the nature of the higher, to grasp within
our ken that which transcends all analogies, because what
we know are merely partial views of it, <h<> purelv
material and physical path affords a means of approach

62 THE FOURTH DIMENSION

pursuing which we are in less likelihood of error than if
we use the more frequently trodden path of framing
conceptions which in their elevation and beauty seem to
us ideally perfect.

For where we are concerned with our own thoughts, the
development of our own ideals, we are as it were on a
curve, moving at any moment in a direction of tangency.
Whither we go, what we set up and exalt as perfect,
represents not the true trend of the curve, but our own
direction at the present — a tendency conditioned by the
past, and by a vital energy of motion essential but
only true when perpetually modified. That eternal cor-
rector of our aspirations and ideals, the material universe
draws sublimely away from the simplest things we can
touch or handle to the infinite depths of starry space,
in one and all uninfluenced by what we think or feel,
presenting unmoved fact to which, think it good or
think it evil, we can but conform, yet out of all that
impassivity with a reference to something beyond our
individual hopes and fears supporting us and giving us
our being.

And to this great being we come with the question :
" You, too, what is your higher ? "

Or to put it in a form which will leave our conclusions in
the shape of no barren formula, and attacking the problem
on its most assailable side : " What is the wheel and the
shafting of the four-dimensional mechanic ? "

In entering on this enquiry we must make a plan of
procedure. The method which I shall adopt is to trace
out the steps of reasoning by which a being confined
to movement in a two-dimensional world could arrive at a
conception of our turning and rotation, and then to apply
an analogous process to the consideration of the higher
movements. The plane being must be imagined as no
abstract figure, but as a real body possessing all three

THE HIGHER WORLD 63

dimensions. His limitation to a plane must be the result
of physical conditions.

We will therefore think of him as of a figure cut out of
paper placed on a smooth plane. Sliding over this plane,
and coming into contact with other figures equally thin
as he in the third dimension, he will apprehend them only
by their edges. To him they will be completely bounded
by lines. A " solid " body will be to him a two-dimensional
extent, the interior of which can only be reached by
penetrating through the bounding lines.

Now such a plane being can think of our three-
dimensional existence in two ways.

First, he can think of it as a series of sections, each like
the solid he knows of extending in a direction unknown
to him, which stretches transverse to his tangible
universe, which lies in a direction at right angles to every
motion which he made.

Secondly, relinquishing the attempt to think of the
three-dimensional solid body in its entirety he can regard
it as consisting of a number of plane sections, each of them
in itself exactly like the two-dimensional bodies he knows,
but extending away from his two-dimensional space.

A square lying in his space he regards as a solid
bounded by four lines, each of which lies in his space.

A square standing at right angles to his plane appears
to him as simply a line in his plane, for all of it except
the line stretches in the third dimension.

He can think of a three-dimensional body as consisting
of a number of such sections, each of which starts from a
line in his space.

Now, since in his world he can make any drawing or
model which involves only 1 wo dimensions, he can represent
each sncli upright section as it actually is, and can repre-
sent a turning from a known into the unknown dimension
as a turning from one to another of his known dimensions.

04 THE FOURTH DIMENSION

To see the whole he must relinquish part of that which
he has, and take the whole portion by portion.

Consider now a plane being in front of a square, fig. 34.

The square can turn about any point

in the plane — say the point A. But it

cannot turn about a line, as AB. For,

in order to turn about the line AB,

the square must leave the plane and

move in the third dimension. This

^ ^ motion is out of his range of observa-

* lg' tion, and is therefore, except for a

process of reasoning, inconceivable to him.

Eotation will therefore be to him rotation about a point.
Eotation about a line will be inconceivable to him.

The result of rotation about a line he can appprehend.
He can see the first and last positions occupied in a half
revolution about the line AC. The result of such a half revo-
lution is to place the square abcd on the left hand instead
of on the right hand of the line AC. It would correspond
to a pulling of the whole body abcd through the line AC,
or to the production of a solid body which was the exact
reflection of it in the line AC. It would be as if the square
abcd turned into its image, the line AB acting as a mirror.
Such a reversal of the positions of the parts of the square
would be impossible in his space. The occurrence of it
would be a proof of the existence of a higher dimensionality.
Let him now, adopting the conception of a three-
dimensional body as a series of
sections lying, each removed a little
farther than the preceding one, in
R direction at right angles to his

plane, regard a cube, fig. 36, as a
series of sections, each like thr

B * square which forms its base, all
Fig. 35. rigidly connected together.

THE HIGHER WORLD

If now lie turns the square about the point A in the
plane of xy, each parallel section turns with the square
he moves. In each of the sections there is a point at
rest, that vertically over a. Hence he would conclude
that in the turning of a three-dimensional body there is
one line which is at rest. That is a three-dimensional
turning in a turning about a line.

In a similar way let us regard ourselves as limited to a
three-dimensional world by a physical condition. Let us
imagine that there is a direction at right angles to every
direction in which we can move, and that we are pre-
vented from passing in this direction by a vast solid, that
against which in every movement wTe make we slip as
the plane being slips against his plane sheet.

We can then consider a four-dimensional body as con-
sisting of a series of sections, each parallel to our space,
and each a little farther off than the preceding on the
unknown dimension.

Take the simplest four-dimensional body — one which

begins as a cube, fig. 3G, in our

space, and consists of sections, each

a cube like fig. 36, lying away from

oar space. If we turn the cube

which is its base in our space

about a line, if, e.g., in fig. 30 we

< urn the cube about the line AH,

not only it but each of t ho parallel

cubes moves about a line. The

cube we Bee moves about the line ah, the cube beyond it

about n lino parallel to ah and soon. Hence the whole

four-dimensional body moves aboul a plane, lor the

e of these linos is our way of thinking aboul the

plain- which, Btaiting from the line \i; in our space, run-
off in t he unknown direct ion.

66 THE FOURTH DIMENSION

In this case all that we see of the plane about which
the turning takes place is the line ah.

But it is obvious that the axis plane may lie in our
space. A point near the plane determines with it a three-
dimensional space. When it begins to rotate round the
plane it does not move anywhere in this three-dimensional
space, but moves out of it. A point can no more rotate
round a plane in three-dimensional space than a point
can move round a line in two-dimensional space.

We will now apply the second of the modes of repre-
sentation to this case of turning about a plane, building
up our analogy step by step from the turning in a plane
about a point and that in space about a line, and so on.

In order to reduce our considerations to those of the
greatest simplicity possible, let us realise how the plane
being would think of the motion by which a square is
turned round a line.

Let, fig. 34, abcd be a square on his plane, and repre-
sent the two dimensions of his space by the axes ax Ay.

Now the motion by which the square is turned over
about the line AC involves the third dimension.

He cannot represent the motion of the whole square in
its turning, but he can represent the motions of parts of
it. Let the third axis perpendicular to the plane of the
paper be called the axis of z. Of the three axes x, y, z,
the plane being can represent any two in his space. Let
him then draw, in fig. 35, two axes, x and z. Here he has
in his plane a representation of what exists in the plane
which goes off perpendicularly to his space.

In this representation the square would not be shown,
for in the plane of xz simply the line ab of the square is
contained.

The plane being then would have before him, in fig. 35,
the representation of one line ab of his square and two
axes, x and z, at right angles. Now it would be obvious

THE HIGHER WOULD 67

to him that, by a turning such as he knows, by a rotation
about a point, the line ab can turn round A, and occu-
pying all the intermediate positions, such as ABi, come
after half a revolution to lie as Ax produced through a.

Again, just as he can represent the vertical plane
through ab, so he can represent the vertical plane
through a'b', fig. 34, and in a like manner can see that
the line a'b' can turn about the point a' till it lies in the
opposite direction from that which it ran in at first.

Now these two turnings are not inconsistent. In his
plane, if AB turned about A, and a'b' about a', the con-
sistency of the square would be destroyed, it would be an
impossible motion for a rigid body to perform. But in
the turning which he studies portion by portion there is
nothing inconsistent. Each line in the square can turn
in this way, hence he would realise the turning of the
whole square as the sum of a number of turnings of
isolated parts. Such turnings, if they took place in his
plane, would be inconsistent, but by virtue of a third
dimension they are consistent, and the result of them all
is that the square turns about the line AC and lies in a
position in which it is the mirror image of what it was in
its first position. Thus he can realise a turning about a
line by relinquishing one of his axes, and representing his
body part by part.

Let u- apply this method to the turning of a cube so as
to become the mirror image of itself. In our space we can
construct three independent axes, x, y, z, shown in fig. 36.
Suppose that there is a fourth axis, wf at right angles to
each and every one of them. We cannot, keeping all
three axes, x, y, z, represent w in our space; but if we
relinquish one of our three axes we can let the fourth axis
take it< place, and we can represent what lies in the
.-pace, determined by the two axes we retain and the
fourth axis.

THE FOURTB DIMENSION

Fier. 37.

Kel us suppose thai we let the y axis drop, and thai

we represent the w axis as occupy-
ing its direction. We have in fig.
37 a drawing of what we should
then see of the cube. The square
abcd, remains unchanged, for that
is in the plane of xz, and we
still have that plane. But from
this plane the cube stretches out
in the direction of the y axis. Now the y axis is gone,
and so we have no more of the cube than the face abcd.

Considering now this face abcd, we
see that it is free to turn about the
line ab. It can rotate in the x to w
direction about this line. In fig. 38
it is shown on its way, and it can
evidently continue this rotation till
it lies on the other side of the z
axis in the plane of xz.
We can also take a section parallel to the face abcd,
and then letting drop all of our space except the plane of
that section, introduce the w axis, running in the old y
direction. This section can be represented by the same
drawing, fig. 38, and we see that it can rotate about the
line on its left until it swings half way round and runs in
the opposite direction to that which it ran in before.
These turnings of the different sections are not incon-
sistent, and taken all together they will bring the cube
from the position shown in fig. 36 to that shown in
fig. 41.

Since wTe have three axes at our disposal in our space,
we are not obliged to represent the w axis by any particular
one. We may let any axis we like disappear, and let the
fourth axis take its place.

In rig. 36 suppose the z axis to go. We have then

Fie. 38.

THE niGHEft WORLD

Fig. 39.

simply the plane of ocy and the square base of the
w- cube aceg, fig. 39, is all that could

be seen of it. Let now the iv axis
take the place of the z axis and
we have, in fig. 39 again, a repre-
sentation of the space of xyiv, in
which all that exists of the cube is
its square base. Now, by a turning
of x to iv, this base can rotate around the line ae, it is

shown on its way in fig. 40, and
finally it will, after half a revolution,
lie on the other side of the y axis.
In a similar way we may rotate
sections parallel to the base of the
xtv rotation, and each of them comes
to run in the opposite direction from
that which they occupied at first.
Thus again the cube comes from the position of fig. 30.

to that of fig. 41. In this x
to iv turning, we see that it
takes place by the rotations of
sections parallel to the front
face about lines parallel to AB,
or else we may consider it as
consisting of the rotation of
sections parallel to the base
about lines parallel to ae. It
is a rotation of the whole cube about the plane abef.
Two separate sections could aot rotate about two separate
lines in our space without conflicting, but their motion is
consistent when we consider another dimension. .Insi,
then, as a plane being can think of rotal ion about a line as
a rotation about a number of points, these rotations not
interfering as they would if they look place in bis two-
dimensional space, bo we can think of a rotation about a

2"? { position

Imposition

Fie. ii.

THE FOUKT11 DIMENSION

plane as ilie rotation of a number of sections of a body
about a number of lines in a plane, these rotations not
1 icing inconsistent in a four-dimensional space as they are
in three-dimensional space.

We are riot limited to any particular direction for the
lines in the plane about which we suppose the rotation
of the particular sections to take place. Let us draw
the section of the cube, fig. 30, through a, F, c, n, forming a
sloping plane. Now since the fourth dimension is at
right angles to every line in our space it is at right
angles to this section also. We can represent our space
by drawing an axis at right angles to the plane ACEG, our
space is then determined by the plane aceg, and the per-
pendicular axis. If we let this axis drop and suppose the
fourth axis, to, to take its place, we have a representation of
the space which runs off in the fourth dimension from the
plane aceg. In this space we shall see simply the section
ACEG of the cube, and nothing else, for one cube does not
extend to any distance in the fourth dimension.

If, keeping this plane, we bring in the fourth dimension,
we shall have a space in which simply this section of
the cube exists and nothing else. The section can turn
about the line af, and parallel sections can turn about

parallel lines. Thus in con-
sidering the rotation about
a plane we can draw any
lines we like and consider
the rotation as taking place
in sections about them.

To bring out this point
more clearly let us take two
parallel lines, a and u, in
the space of xyz, and let CD
lg" ' and ef be two rods running

above and below the plane of xy, from these lines. If we

THE HIGHER WORLD

turn these rods in our space about the lines A and B, as
the upper end of oue, F, is going down, the lower end of
the other, c, will be coming up. They will meet and
conflict. But it is quite possible for these two rods
each of them to turn about the two lines without altering
their relative distances.

To see this suppose the y axis to go, and let the iv axis
take its place. We shall see the lines A and B no longer,
for they run in the y direction from the points G and H.
Fig. 43 is a picture of the two rods seen in the space

of xztu. If they rotate in the
direction shown by the arrows —
in the z to w direction — they
move parallel to one another,
keeping their relative distances.
Each will rotate about its own
line, but their rotation will not
be inconsistent with their form-
ing part of a rigid body.

Now we have but to suppose
a central plane with rods crossing
it at every point, like CD and ef cross the plane of xy,
to have an image of a mass of matter extending equal
diMances on each side of a diametral plane. As two of
these rods can rotate round, so can all, and the whole
mass of matter can rotate round its diametral plane.

This rotation round a plane corresponds, in four
dimensions, to the rotation round an axis in three
dimensions. Rotation of a body round a plane is the
analogue of rotation of a rod round an axis.

Tn a plane we have rotation round a point, in three-
space rotation round an axis line, in four-space rotation
round an axis plane

The four-dimensional being's sliafi by which lie trans-
mits power is a disk rotating round its central plane — ■

t .w

I THE FOURTH DIMENSION

the whole contour corresponds to the ends of an axis
of rotation in our space. He can imparl the rotation at

an v point and take it off at any other point on the contour,
just as rotation round a line can in three-space be imparted

at one end of a rod and taken off at the other end.

A four-dimensional wheel can easily be described from
the analogy of the representation which a plane being
would form for himself of one of our wheels.

Suppose a wheel to move transverse to a plane, so that
the whole disk, which I will consider to be solid and
without spokes, came at the same time into contact with
the plane. It would appear as a circular portion of plane
matter completely enclosing another and smaller portion —
the axle.

This appearance would last, supposing the motion of
the wheel to continue until it had traversed the plane by
the extent of its thickness, when there would remain in
the plane only the small disk which is the section of the
axle. There would be no means obvious in the plane
at first by which the axle could be reached, except by
going through the substance of the wheel. But the
possibility of reaching it without destroying the substance
of the wheel would be shown by the continued existence
of the axle section after that of the wheel had disappeared.

In a similar way a four-dimensional wheel moving
transverse to our space would appear first as a solid sphere,
completely surrounding a smaller solid sphere. The
outer sphere would represent the wheel, and would last
until the wheel has traversed our space by a distance
equal to its thickness. Then the small sphere alone
would remain, representing the section of the axle. The
large sphere could move round the small one quite freely.
Any line in space could be taken as an axis, and round
this line the outer sphere could rotate, while the inner
sphere remained still. But in all these directions of

THE HIGHER WORLD 73

revolution there would be in reality one line which
remained unaltered, that is the line which stretches away
in the fourth direction, forming the axis of the axle. The
four-dimensional wheel can rotate in any number of planes,
but all these planes are such that there is a line at right
angles to them all unaffected by rotation in them.

An objection is sometimes experienced as to this mode
of reasoning from a plane world to a higher dimensionality.
How artificial, it is argued, this conception of a plane
world is. If any real existence confined to a superficies
could be shown to exist, there would be an argument for
one relative to which our three-dimensional existence is
superficial. But, both on the one side and the other of
the space we are familiar with, spaces either with less
or more than three dimensions are merely arbitrary
conceptions.

In reply to this I would remark that a plane being
having one less dimension than our three would have one-
third of our possibilities of motion, while we have only
one-fourth less than those of the higher space. It may
very well be that there may be a certain amount of
freedom of motion which is demanded as a condition of an
organised existence, and that no material existence is
aible with a more limited dimensionality than ours.
This is well seen if we try to construct the mechanics of a
two-dimensional world. No tube could exist, for unless
joined together completely at one end two parallel lines
would be completely separate. The possibility of an
organic structure, Bubjeel to conditions such as this, is
highly problematical; yet, possibly in the convolutions
of the brain there may be a mode of existence to be
described as two-dimensional.

We have but to suppose the increase in surface and
the diminution in mass carried on to a certain extent
to find a region which, though without mobility of (he

74 THE FOUllTH DIMENSION

constituents, would have to be described as two-dimensional.

But, however artificial the conception of a plane being
may be, it is none the less to be used in passing to the
conception of a greater dimensionality than ours, and
hence the validity of the first part of this objection
altogether disappears directly we find evidence for such a
state of being.

The second part of the objection has more weight.
How is it possible to conceive that in a four-dimensional
space any creatures should be confined to a three-
dimensional existence?

In reply I would say that we know as a matter of fact
that life is essentially a phenomenon of surface. The
amplitude of the movements which we can make is much
greater along the surface of the earth than it is up
or down.

Now we have but to conceive the extent of a solid
surface increased, while the motions possible tranverse to
it are diminished in the same proportion, to obtain the
image of a three-dimensional world in four-dimensional
space.

And as our habitat is the meeting of air and earth on
the world, so we must think of the meeting place of two
as affording the condition for our universe. The meeting
of what two ? What can that vastness be in the higher
space which stretches in such a perfect level that our
astronomical observations fail to detect the slightest
curvature ?

The perfection of the level suggests a liquid — a lake
amidst what vast scenery ! — whereon the matter of Hi«
universe floats speck-like.

But this aspect of the problem is like what are called
in mathematics boundary conditions.

We can trace out all the consequences of four-dimen-
sional movements down to their last detail. Then, knowing

TEE filGflER WotiLD. 75

the mode of action which would be characteristic of the
minutest particles, if they were free, we can draw con-
clusions from what they actually do of what the constraint
on them is. Of the two things, the material conditions and
the motion, one is known, and the other can be inferred.
If the place of this universe is a meeting of two, there
would be a one-sideness to space. If it lies so that what
stretches away in one direction in the unknown is unlike
what stretches away in the other, then, as far as the
movements which participate in that dimension are con-
cerned, there would be a difference as to which way the
motion took place. This would be shown in the dissimi-
larity of phenomena, which, so far as all three-space
movements are concerned, were perfectly symmetrical.
To take an instance, merely, for the sake of precising
our ideas, not for any inherent probability in it ; if it could
be shown that the electric current in the positive direction
were exactly like the electric current in the negative
direction, except for a reversal of the components of the
motion in three-dimensional space, then the dissimilarity
of the discharge from the positive and negative poles
would be an indication of a one-sideness to our space.
The only cause of difference in the two discharges would
be due to a component in the fourth dimension, which
directed in one direction transverse to our space, met with
a different resistance to that which it met when directed
in the opposite direction.

CHAPTER VII

THE EVIDENCES FOR A FOURTH DIMENSION

The method necessarily to be employed in the search for
the evidences of a fourth dimension, consists primarily in
the formation of the conceptions of four-dimensional
shapes and motions. When we are in possession of these
it is possible to call in the aid of observation, without
them we may have been all our lives in the familiar
presence of a four-dimensional phenomenon without ever
recognising its nature.

To take one of the conceptions we have already formed,
the turning of a real thing into its mirror image would be
an occurrence which it would be hard to explain, except on
the assumption of a fourth dimension.

We know of no such turning. But there exist a multi-
tude of forms which show a certain relation to a plane,
a relation of symmetry, which indicates more than an acci-
dental juxtaposition of parts. In organic life the universal
type is of right- and left-handed symmetry, there is a plain1
on each side of which the parts correspond. Now we have
seen that in four dimensions a plane takes the place of a
line in three dimensions. In our space, rotation about an
axis is the type of rotation, and the origin of bodies sym-
metrical about a line as the earth is symmetrical about an
axis can easily be explained. But where there is symmetry

about a plane no simple physical motion, such as we

THE EVIDENCES FOE A FOURTH DIMENSION" 77

are accustomed to, suffices to explain it. In our space a
symmetrical object must be built up by equal additions
on each side of a central plane. Such additions about
such a plane are as little likely as any other increments.
The probability against the existence of symmetrical
form in inorganic nature is overwhelming in our space,
and in organic forms they would be as difficult of produc-
tion as any other variety of configuration. To illustrate
this point we may take the child's amusement of making
from dots of ink on a piece of paper a life-like repre-
sentation of an insect by simply folding the paper
over. The dots spread out on a symmetrical line, and
give the impression of a segmented form with antenna*
and legs.

Now seeing a number of such figures we should
naturally infer a folding over. Can, then, a folding over
in four-dimensional space account for the symmetry of
organic forms ? The folding cannot of course be of the
bodies we see, but it may be of those minute constituents,
1 lie ultimate elements of living matter which, tinned in one
way or the other, become right- or left-handed, and so
produce a corresponding structure.

There is something in life not included in our concep-
tions of mechanical movement. Is this something a four-
dimensional movement?

If we look at it from the broadest point of view, there is
something striking in the fact that where life conies in
there arises an entirely different set of phenomena to
those of the inorganic world.

The interest and values of life as we know it in our-
selves, as we know il existing around us in subordinate
forms, is entirely and completely different to anything
which inorganic nature shows. And in living beings we
have a kind of form, a disposition <>l matter which is
entirely different from that shown in inorganic matter.

78 THE FOURTH DIMENSION

Right- and left-banded symmetry does not occur in the
configurations of dead matter. We have instances of
symmetry about an axis, but not about a plane. It can
be argued that the occurrence of symmetry in two dimen-
sions involves the existence of a three-dimensional process,
as when a stone falls into water and makes rings of ripples,
or as when a mass of soft material rotates about an axis.
It can be argued that symmetry in any number of dimen-
sions is the evidence of an action in a higher dimensionality.
Thus considering living beings, there is an evidence both
in their structure, and their different mode of activity, of a
something coming in from without into the inorganic
world.

And the objections which will readily occur, such as
those derived from the forms of twin crystals and the
theoretical structure of chemical molecules, do not in-
validate the argument ; for in these forms too the
presumable seat of the activity producing them lies in that
very minute region in which we necessarily place the seat
of a four-dimensional mobility.

In another respect also the existence of symmetrical forms
is noteworthy. It is puzzling to conceive how two shapes
exactly equal can exist which are not superposible. Such
a pair of symmetrical figures as the two hands, right and
left, show either a limitation in our power of movement,
by which we cannot superpose the one on the other, or a
definite influence and compulsion of space on matter,
inflicting limitations which are additional to those of the
proportions of the parts.

We will, however, put aside the arguments to be drawn
from the consideration of symmetry as inconclusive,
retaining one valuable indication which they afford. If
it is in virtue of a four-dimensional motion that sym-
metry exists, it is only in the very minute particles
of bodies that that motion is to be found, for there is

THE EVIDENCES FOR A FOURTH DIMENSION 79

no such thing as a bending over in four dimensions of
any object of a size which we can observe. The region
of the extremely minute is the one, then, which we
shall have to investigate. We must look for some
phenomenon which, occasioning movements of the kind
we know, still is itself inexplicable as any form of motion
which we know.

Now in the theories of the actions of the minute
particles of bodies on one another, and in the motions of
the ether, mathematicians have tacitly assumed that the
mechanical principles are the same as those which prevail
in the case of bodies which can be observed, it has been
assumed without proof that the conception of motion being
three-dimensional, holds beyond the region from observa-
tions in which it was formed.

Hence it is not from any phenomenon explained by
mathematics that we can derive a proof of four dimensions.
Every phenomenon that has been explained is explained
as three-dimensional. And, moreover, since in the region
of the very minute we do not find rigid bodies acting
on each other at a distance, but elastic substances and
continuous fluids such as ether, we shall have a double
task.

We must form the conceptions of the possible move-
ments of elastic and liquid four-dimensional matter, before
we can begin to observe. Let us, th erefore, take the four-
dimensional rotation about a plane, and enquire what it
becomes in the case of extensible fluid substances. If
four-dimensional movements exist, this kind of rotation
must exist, and the finer portions of matter must exhibit
it.

Consider for a moment a rod of flexible and extensible
material. It can turn aboutan axis, even if not straight ;
a ring of iudia rubber can turn inside out.

What would t his be in t he ease of four dimensions ?

THE FOURTH IHMKNSION

Fig. 44.
Axis ofx running towards
the observer.

V 3

us consider a sphere of our three-dimensional

matter having a definite
thickness. To represent
this thickness let us sup-
pose that from every point
of the sphere in fig. 44 rods
/ \ project both ways, in and

IJ out, like D and F. We can
only see the external por-
tion, because the internal
parts are hidden by the
sphere.

In this sphere the axis
of x is supposed to come
towards the observer, the
axis of z to run up, the axis of y to go to the right.

Now take the section determined by the zy plane.

This will be a circle as
shown in fig. 45. If we
let drop the x axis, this
circle is all we have of
the sphere. Letting the
iv axis now run in the
place of the old x axis
we have the space yzw,
and in this space all that
we have of the sphere is
the circle. Fig. 45 then
represents all that there
is of the sphere in the
space of yziv. In this space it is evident that the rods
CD and EF can turn round the circumference as an axis.
If the matter of the spherical shell is sufficiently exten-
sible to allow the particles C and E to become as widely
separated as they would be in the positions D and F, then

Fiff. 45.

THE EVIDENCES FOR A FOURTH DIMENSION 81

the strip of matter represented by CD and ef and a
multitude of rods like them can turn round the circular
circumference.

Thus this particular section of the sphere can turn
inside out, and what holds for any one section holds for
all. Hence in four dimensions the whole sphere can, if
extensible turn inside out. Moreover, any part of it —
a bowl-shaped portion, for instance — can turn inside out,
and so on round and round.

This is really no more than we had before in the
rotation about a plane, except that we see that the plane
can, in the case of extensible matter, be curved, and still
play the part of an axis.

If we suppose the spherical shell to be of four-dimen-
sional matter, our representation will be a little different.
Let us suppose there to be a small thickness to the matter
in the fourth dimension. This would make no difference
in fig. 44, for that merely shows the view in the xyz
space. But when the x axis is let drop, and the w axis
comes in, then the rods CD and ef which represent the
matter of the shell, will have a certain thickness perpen-
dicular to the plane of the paper on which they are drawn.
If they have a thickness in the fourth dimension they will
show this thickness when looked at from the direction of
the w axis.

Supposing these rods, then, to be small slabs strung on
the circumference of the circle in fig. 45, we see that
there will not be in this case either any obstacle to their
1 inning round the circumference. We can have a shell
of extensible material or of fluid material turning inside
<»uf in four dimensions.

And we must remember that in four dimensions there
i- no Blich thing as rotation round an axis. If we want to
investigate the motion of fluids in four dimensions we
must take a movement about an axis in our space, and

82 THE FOURTH DIMENSION

find the corresponding movement about a plane in
four space.

Now, of all the movements which take place in fluids,
the most important from a physical point of view is
vortex motion.

A vortex is a whirl or eddy — it is shown in the gyrating
wnaths of dust seen on a summer day; it is exhibited on
a larger scale in the destructive march of a cyclone.

A wheel whirling round will throw off the water on it.
But when this circling motion takes place in a liquid
itself it is strangely persistent. There is, of course, a
certain cohesion between the particles of water by which
they mutually impede their motions. But in a licpaid
devoid of friction, such that every particle is free from
lateral cohesion on its path of motion, it can be shown
that a vortex or eddy separates from the mass of the
fluid a certain portion, which always remain in that
vortex.

The shape of the vortex may alter, but it always con-
sists of the same particles of the fluid.

Now, a very remarkable fact about such a vortex is that
the ends of the vortex cannot remain suspended and
isolated in the fluid. They must always run to the
boundary of the fluid. An eddy in water that remains
half way down without coming to the top is impossible.

The ends of a vortex must reach the boundary of a
fluid — the boundary may be external or internal — a vortex
may exist between two objects in the fluid, terminating
one end on each object, the objects being internal
boundaries of the fluid. Again, a vortex may have its
ends linked together, so that it forms a ring. Circular
vortex rings of this description are often seen in puffs of
smoke, and that t lie smoke travels on in the ring is a
proof that the vortex always consists of the same particles
of air.

THE EVIDENCES FOR A FOURTH DIMENSION 83

Let us now enquire what a vortex would be in a four-
dimensional fluid.

We must replace the line axis by a plane axis. We
should have therefore a portion of fluid rotating round
a plane.

We have seen that the contour of this plane corresponds
with the ends of the axis line. Hence such a four-
dimensional vortex must have its rim on a boundary of
the fluid. There would be a region of vorticity with a
contour. If such a rotation were started at one part of a
circular boundary, its edges would run round the boundary
in both directions till the whole interior region was filled
with the vortex sheet.

A vortex in a three-dimensional liquid may consist of a
number of vortex filaments lying together producing a
tube, or rod of vorticity.

In the same way we can have in four dimensions a
number of vortex sheets alongside each other, each of which
can be thought of as a bowl-shaped portion of a spherical
shell turning inside out. The rotation takes place at any
point not in the space occupied by the shell, but from
that space to the fourth dimension and round back again.

Is there anything analogous to this within the range
of our observation ?

An electric current answers this description in every
respect. Electricity does not flow through a wire. Its eft ret
travels both ways from the starting point along the wire.
The spark which shows its passing midway in its circuit
is later than that which occurs at points near its starting
point on either side of it.

Moreover, it is known thai the action of the current
Is not in the wire. It is in the region enclosed by the
wire, this is the field of force, the Locus of the exhibition
of t he < fifed - of t he current .

And the necessity of a conduct Lng circuit for a current is

84 THE FOUftTH DIMENSION

exactly that which we should expect if it were a four-dimen-
sional vortex. According to Maxwell every current forma
a closed circuit, and this, from the four-dimensional point
of view, is the same as saying a vortex must have its ends
on a boundary of the fluid.

Thus, on the hypothesis of a fourth dimension, the rota-
tion of the fluid ether would give the phenomenon of an
electric current. We must suppose the ether to be full of
movement, for the more we examine into the conditions
which prevail in the obscurity of the minute, the more we
find that an unceasing and perpetual motion reigns. Thus
wre may say that the conception of the fourth dimension
means that there must be a phenomenon which presents
the characteristics of electricity.

We know now that light is an electro-magnetic action,
and that so far from being a special and isolated pheno-
menon this electric action is universal in the realm of the
minute. Hence, may wTe not conclude that, so far from
the fourth dimension being remote and far away, being a
thing of symbolic import, a term for the explanation of
dubious facts by a more obscure theory, it is really the
most important fact within our knowledge. Our three-
dimensional world is superficial. These processes, which
really lie at the basis of all phenomena of matter,
escape our observation by their minuteness, but reveal
to our intellect an amplitude of motion surpassing any
that we can see. In such shapes and motions there is a
realm of the utmost intellectual beauty, and one to
which our symbolic methods apply with a better grace
than they do to tho^e of three dimensions.

CHAPTER VIII

THE USE OF FOUR DIMENSIONS IN

THOUGHT

Having held before ourselves this outline of a conjecture
of the world as four-dimensional, having roughly thrown
together those facts of movement which we can see apply
to our actual experience, let us pass to another branch
of our subject.

The engineer uses drawings, graphical constructions,
in a variety of manners. He has, for instance, diagrams
which represent the expansion of steam, the efficiency
of his valves. These exist alongside the actual plans of
his machines. They are not the pictures of anything
really existing, but enable him to think about the relations
which exist in his mechanisms.

And so, besides showing us the actual existence of that
world which lies beneath the one of visible movements,
four-dimensional space enables us to make ideal con-
structions which serve to represent the relations of tilings,
and throw what would otherwise be obscure into a definite
and Buggesl ive form.

Prom amidst the great variety of instances which lies
before me I will select two, one dealing with a subject
ol Blight intrinsic interest, which however gives within
;i limited field a gt riking example of the methocj

86 THE FOURTH DIMENSION

of drawing conclusions and the use of higher space
figures.*

The other instance is chosen on account of the bearing
it has on our fundamental conceptions. In it I try to
discover the real meaning of Kant's theory of experience.

The investigation of the properties of numbers is much
facilitated by the fact that relations between numbers are
themselves able to be represented as numbers — e.g., 12,
and 3 are both numbers, and the relation between them
is 4, another number. The way is thus opened for a
process of constructive theory, without there being any
necessity for a recourse to another class of concepts
besides that which is given in the phenomena to be
studied.

The discipline of number thus created is of great and
varied applicability, but it is not solely as quantitative
that wTe learn to understand the phenomena of nature.
It is not possible to explain the properties of matter
by number simply, but all the activities of matter are
energies in space. They are numerically definite and also,
we may say, directedly definite, i.e. definite in direction.

Is there, then, a body of doctrine about space which, like
that of number, is available in science ? It is needless
to answer : Yes ; geometry. But there is a method
lying alongside the ordinary methods of geometry, which
tacitly used and presenting an analogy to the method
of numerical thought deserves to be brought into greater
prominence than it usually occupies.

The relation of numbers is a number.

Can we say in the same way that the relation of
shapes is a shape ?

We can.

* It is suggestive also in another respect, because it shows very
clearly that in our processes of thought there are in play faculties other
than logical ; in it the origin of the. idea which proves to be justified is
'li.r.vn from t lie eon-idi ration of symmetry, a branch of the beautiful,

THE USE OF FOUR DIMENSIONS IN THOUGHT 87

To take an instance chosen on account of its ready

availability. Let us take

two right-angled triangles of

a given hvpothenuse, but

] laving sides of different

W* 4G- lengths (fig. 40). These

triangles are shapes which have a certain relation to each

other. Let us exhibit their relation as a figure.

Draw two straight lines at right angles to each other,
vl * the one HL a horizontal level, the

other VL a vertical level (fig. 47).
By means of these two co-ordin-
ating lines we can represent a
double set of magnitudes ; one set
j as distances to the right of the ver-

Fig- 4~- tical level, the other as distances

above the horizontal level, a suitable unit being chosen.

Thus the line marked 7 will pick out the assemblage
of points whose distance from the vertical level is 7,
and the line marked 1 will pick out the points wdiose
distance above the horizontal level is 1. The meeting
point of these two lines, 7 and 1, will define a point
which with regard to the one set of magnitudes is 7,
with regard to the other is 1. Let us take the sides of
our triangles as the two sets of magnitudes in <|nostion.
Then the point 7, 1, will represent the triangle whose
Bides are 7 and 1. Similarly the point 5, 5 — 5, that
i-. to the right of (lie vertical level and 5 above the
^5,5 horizontal level — will represent (he

triangle whose sides are 5 and 5
(fig. 48).

Thus we have obtained a figure
consist ing of I lie t wo points 7, 1 ,
"8 and 5, 5, representative <>(' our two

triangles, lint we can go further, and, drawing an arc

88 THl FOURTH DIMENSION

of a circle about o, the meeting point of the horizont.il
and vertical levels, which passes through 7, 1, and o, 5,
assert that all the triangles which are right-angled and
have a hypothenuse whose square is 50 are represented
by the points on this arc.

Thus, each individual of a class being represented by a
point, the whole class is represented by an assemblage of
points forming a figure. Accepting this representation
we can attach a definite and calculable significance to the
expression, resemblance, or similarity between two indi-
viduals of the class represented, the difference being
measured by the length of the line between two repre-
sentative points. It is needless to multiply examples, or
to show how, corresponding to different classes of triangles,
we obtain different curves.

A representation of this kind in which an object, a
thing in space, is represented as a point, and all its pro-
perties are left out, their effect remaining only in the
relative position which the representative point bears
to the representative points of the other objects, may be
called, after the analogy of Sir William Hamilton's
hodograph, a " Poiograph."

Eepresentations thus made have the character of
natural objects; they have a determinate and definite
character of their own. Any lack of completeness in them
is probably due to a failure in point of completeness
of those observations which form the ground of their
construction.

Every system of classification is a poiograph. In
Mendeleeff's scheme of the elements, for instance, each
element is represented by a point, and the relations
between the elements are represented by the relations
between the points.

So far I have simply brought into prominence processes
and considerations with which we are all familiar. But

THE USE OF FOUR DIMENSIONS IN THOUGHT 89

it is worth while to bring into the full light of our atten-
tion our habitual assumptions and processes. It often
happens that we find there are two of them which have
a bearing on each other, which, without this dragging into
the light, we should have allowed to remain without
mutual influence.

There is a fact which it concerns us to take into account
in discussing the theory of the poiograph.

With respect to our knowledge of the world we are
far from that condition which Laplace imagined when he
asserted that an all-knowing mind could determine the
future condition of every object, if he knew the co-ordinates
of its particles in space, and their velocity at any
particular moment.

On the contrary, in the presence of any natural object,
we have a great complexity of conditions before us,
which we cannot reduce to position in space and date
in time.

There is mass, attraction apparently spontaneous, elec-
trical and magnetic properties which must be superadded
to spatial configuration. To cut the list short we must
say that practically the phenomena of the world present
us problems involving many variables, which we must
take as independent.

From this it follows that in making poiographs we
must be prepared to use space of more than three dimen-
sions. If the symmetry and completeness of our repre-
sentatation is to be of use to us we must be prepared to
appreciate and criticise figures of a complexity greater
t lian of t hose in three dimensions. It is impossible to give
an example of such a poiograph which will not be merely
trivial, without going into details of some kind irrelevant
to our subject. I prefer to introduce the irrelevant details
rather than treat this part of the subject perfunctorily.

To take an instance of a poiograph which does not lead

THE FOUTlTn DIMENSION

us into the complexities incident on its application in
classificatory science, let us follow Mrs. Alicia Boole Stott
in her representation of the syllogism by its means. She
will be interested to find that the curious gap she detected
has a significance.

A syllogism consists of two statements, the major and
the minor premiss, with the conclusion that can be drawn
from them. Thus, to take an instance, fig. 49. It is
evident, from looking at the successive figures that, if we
know that the region M lies altogether within the region
P, and also know that the region s lies altogether within
the region M, we can conclude that the region s lies
altogether within the region p. M is P,
major premiss ; s is M, minor premiss ; s
is p, conclusion. Given the first two data
we must conclude that s lies in p. The
conclusion s is P involves two terms, s and
p, which are respectively called the subject
and the predicate, the letters s and P
being chosen with reference to the parts
the notions they designate play in the
conclusion, s is the subject of the con-
clusion, P is the predicate of the conclusion.
The major premiss we take to be, that
which does not involve s, and here we
always write it first.

There are several varieties of statement
possessing different degrees of universality and manners of
assertiveness. These different forms of statement are
called the moods.

We will take the major premiss as one variable, as a
thing capable of different modifications of the same kind,
the minor premiss as another, and the different moods we
will consider as defining the variations which these
variables undergo,

Fig. 49.

THE USE OF FOUR DIMENSIONS IN THOUGHT

There are four moods : —

1 . The universal affirmative ; all M is r, called mood A.

2. The universal negative ; no M is P, mood E.

3. The particular affirmative ; some M is r, mood I.

4. The particular negative ; some m is not P, mood o.
The dotted lines in 3 and 4, fig. 50, denote that it is

not known whether or no any objects exist, corresponding

Mood E.

3.
Mood I.

Mood o.

Fie:. 50.

to the space of which the dotted line forms one delimiting

boundary ; thus, in mood I we do not know if there are

any m's which are not P, we only know some m's are p.

Representing the first premiss in its various moods by

q regions marked by vertical lines to

the right of pq, we have in fig. 51,

running up from the four letters Alio,

four columns, each of which indicates

thai the major premiss is in the mood

denoted by the respective letter. In

the first column to the right of pq is

» o

p a rr

Kg. 51,

the mood a. Now above the Line i;s let there be marked
off four regions corresponding to the four moods of the
minor premiss. Thus, in the first row above rs all the
region between us and the firsl horizontal Line above it
denotes thai the minor premiss ia in the mood a. The

THE FOrilTIl DIMENSION

letters e, I, o, in the same way show the mood character-
ising the minor premiss in the rows opposite these letters.
We have still to exhibit the conclusion. To do this we
must consider the conclusion as a third variable, character-
ised in its different varieties by four moods — this being
the syllogistic classification. The introduction of a third
variable involves a change in our system of representation.
Before we started with the regions to the right of a
certain line as representing successively the major premiss
in its moods ; now we must start with the regions to the
right of a certain plane. Let LMNR
be the plane face of a cube, fig. 52, and
let the cube be divided into four parts
by vertical sections parallel to LMNR.
The variable, the major premiss, is re-
presented by the successive regions
Fig. 52. which occur to the right of the plane

LMNR — that region to which A stands opposite, that
slice of the cube, is significative of the mood A. This
whole quarter-part of the cube represents that for even-
part of it the major premiss is in the mood A.

In a similar manner the next section, the second with
the letter E opposite it, represents that for every one of
the sixteen small cubic spaces in it, the major premiss is
in the mood E. The third and fourth compartments made
by the vertical sections denote the major premiss in the
moods i and o. But the cube can be divided in other
ways by other planes. Let the divisions, of which four
stretch from the front face, correspond to the minor
premiss. The first wall of sixteen cubes, facing the
observer, has as its characteristic that in each of the small
cubes, whatever else may be the case, the minor premiss is
in the mood A. The variable — the minor premiss — varies
through the phases A, E, I, o, away from the front face of the
cube, or the front plane of which the front face is a part.

The use of four dimensions In thought 93

And now we can represent the third variable in a precisely
similar way. We can take the conclusion as the third
variable, going through its four phases from the ground
plane upwards. Each of the small cubes at the base of
the whole cube has this true about it, whatever else may
be the case, that the conclusion is, in it, in the mood A.
Thus, to recapitulate, the first wall of sixteen small cubes,
the first of the four walls which, proceeding from left to
right, build up the whole cube, is characterised in each
part of it by this, that the major premiss is in the mood A.

The next wall denotes that the major premiss is in the
mood E, and so on. Proceeding from the front to the
back the first wall presents a region in every part of
which the minor premiss is in the mood A. The second
wall is a region throughout which the minor premiss is in
the mood E, and so on. In the layers, from the bottom
upwards, the conclusion goes through its various moods
beginning with A in the lowest, E in the second, I in the
third, o in the fourth.

In the general case, in which the variables represented
in the poiograph pass through a wide range of values, the
planes from which we measure their degrees of variation
in our representation are taken to be indefinitely extended.
In this case, however, all we are concerned with is the
finite region.

We have now to represent, by some limitation of the
complex we have obtained, the fact that not every com-
bination of premisses justifies any kind of conclusion.
This can be simply effected by marking the regions in
which the premisses, being such as are defined by the
positions, a conclusion which is valid is found.

Taking the conjunction of the major premiss, all M is
l'. and I he minor, all s is m. We conclude that all s is p,
Hence, that region must be marked in which we have the
conjunction of major premiss in mood a ; minor premiss,

THE FOURTH DIMENSION

rf^jgg^

mood A ; conclusion, mood A. This is the cube occupying
the lowest left-hand corner of the large cube.

Proceeding in this way, we find that the regions which
must be marked are those shown in fig. 53.
To discuss the case shown in the marked
cube which appears at the top of fig. ."j::.
Here the major premiss is in the second
wall to the right — it is in the mood E and
is of the type no M is p. The minor
premiss is in the mood characterised by
the third wall from the front. It is of
the type some s is m. From these premisses we draw
the conclusion that some s is not r, a conclusion in the
mood o. Now the mood o of the conclusion is represented
in the top layer. Hence we see that the marking is
correct in this respect.

It would, of course, be possible to represent the cube on

a plane by means of four

squares, as in fig. 54, if we
consider each square to re-
present merely the beginning
of the region it stands for.
Thus the whole cube can be
represented by four vertical
squares, each standing for a
kind of vertical tray, and the
markings would be as shown. In No. 1 the major premiss
is in mood A for the whole of the region indicated by the
vertical square of sixteen divisions j in No. 2 it is in the
mood E, and so on.

A creature confined to a plane would have to adopt some
such disjunctive way of representing the whole cube. He
would be obliged to represent that which we see as a
whole in separate parts, and each part would merely
represent^ would not be, that solid content which we see.

Fig. 54.

THE USE OF FOUR DIMENSIONS IN THOUGHT 95

The view of these four squares which the plane creature
would have would not be such as ours. He would not
see the interior of the four squares represented above, but
each would be entirely contained within its outline, the
internal boundaries of the separate small squares he could
not see except by removing the outer squares.

We are now ready to introduce the fourth variable
involved in the syllogism.

In assigning letters to denote the terms of the syllogism
we have taken s and P to represent the subject and
predicate in the conclusion, and thus in the conclusion
their order is invariable. But in the premisses we have
taken arbitrarily the order all M is p, and all s is M.
There is no reason why M instead of P should not be the
predicate of the major premiss, and so on.

Accordingly we take the order of the terms in the pre-
misses as the fourth variable. Of this order there are four
varieties, and these varieties are called figures.

Using the order in which the letters are written to
denote that the letter first written is subject, the one
written second is predicate, we have the following pos-
sibilities : —

1st Figure.

Major M p

2nd Figure.
P M

3rd Figure.
M P

4th Figure.
P M

Minor .s M

S M

M S

There are therefore four possibilities with regard to
this fourth variable as with regard to the premisses.

We have used up our dimensions of space in represent-
ing the phases of the premisses and the conclusion in
respect of mood, and to represent in an analogous manner
the variations in figure we require a fourth dimension.

New in bringing in Oil's fourth dimension we must
make a change in our origins of measurement analogous
to that which we made in passing from the plane to the
solid.

UO THE FOURTH DIMENSION

This fourth dimension is supposed to run at right

angles to any of the three space dimensions, as the third
space dimension runs at right angles to the two dimen-
sions of a plane, and thus it gives us the opportunity of
generating a new kind of volume. If the whole cube
moves in this dimension, the solid itself traces out a path,
each section of which, made at right angles to the
direction in which it moves, is a solid, an exact repetition
of the cube itself.

The cube as we see it is the beginning of a solid of such
a kind. It represents a kind of tray, as the square face of
the cube is a kind of tray against which the cube rests.

Suppose the cube to move in this fourth dimension in
four stages, and let the hyper-solid region traced out in
the first stage of its progress be characterised by this, that
the terms of the syllogism are in the first figure, then we
can represent in each of the three subsequent stages the
remaining three figures. Thus the whole cube forms
the basis from which we measure the variation in figure.
The first figure holds good for the cube as we see it, and
for that hyper-solid which lies within the first stage ;
the second figure holds good in the second stage, and
so on.

Thus we measure from the whole cube as far as figures
are concerned.

But we saw that when we measured in the cube itself
having three variables, namely, the two premisses and
the conclusion, we measured from three planes. The base
from which we measured was in every case the same.

Hence, in measuring in this higher space we should
have bases of the same kind to measure from, we should
have solid bases.

The first solid base is easily seen, it is the cube itself.
The other can be found from this consideration.

That solid from which we measure figure is that in

THE USE OF FOUR DIMENSIONS IN THOUGHT 07

which the remaining variables run through their full
range of varieties.

Now, if we want to measure in respect of the moods of
the major premiss, we must let the minor premiss, the
conclusion, run through their range, and also the order
of the terms. That is we must take as basis of measure-
ment in respect to the moods of the major that, which
represents the variation of the moods of the minor, the
conclusion and the variation of the figures.

Now the variation of the moods of the minor and of the
conclusion are represented in the square face on the left
of the cube. Here are all varieties of the minor premiss
and the conclusion. The varieties of the figures are
represented by stages in a motion proceeding at right
angles to all space directions, at right angles consequently
to the face in question, the left-hand nice of the cube.

Consequently letting the left-hand face move in this
direction we get a cube, and in this cube all the varieties
of the minor premiss, the conclusion, and the figure are
represented.

Thus another cubic base of measurement is given to
the cube, generated by movement of the left-hand square
in t lie fourth dimension.

We find the other bases in a similar manner, one is the
cube generated by the front square moved in the fourth
dimension so as to generate a cube. From tin's cube
variations in the mood of the minor are measured. The
fourth base is that found by moving the bottom square of
the cube in the fourth dimension. In this cube the
variations of the major, the minor, and t he figure are given.
Considering this as a basis in the four stages proceeding
from it, the variation in the moods of the conclusion are
given.

Any one of i hese cubic bases can be represented in space,
and then the higher solid generated from them lies oul of

'.'^ THE FOURTH DIMENSION

our space. It can only be represented by a device analogous
to thai by which the plane being represents a cube.

He represents the cube shown above, by taking four
square sections and placing them arbitrarily at convenient
distances the one from the other.

So we must represent this higher solid by four cubes :
each cube represents only the beginning of the correspond-
ing higher volume.

It is sufficient for us, then, if we draw four cubes, the
first representing that region in which the figure is of the
first kind, the second that region in which the figure is
of the second kind, and so on. These cubes are the
beginnings merely of the respective regions — they are
the trays, as it were, against which the real solids must
be conceived as resting, from which they start. The first
one, as it is the beginning of the region of the first figure,
is characterised by the order of the terms in the premises
being that of the first figure. The second similarly has
the terms of the premisses in the order of the second
figure, and so on.

These cubes are shown below.

For the sake of showing the properties of the method
of representation, not for the logical problem, I will make
a digression. I will represent in space the moods of the
minor and of the conclusion and the different figures,
keeping the major always in mood A. Here we have
three variables in different stages, the minor, the con-
clusion, and the figure. Let the square of the left-hand
side of the original cube be imagined to be standing by
itself, without the solid part of the cube, represented by
(2) fig. 55. The A, E, I, o, which run away represent the
moods of the minor, the A, E, I, o, which run up represent
the moods of the conclusion. The whole square, since it
is the beginning of the region in the major premiss, mood
A, is to be considered as in major premiss, mood A.

THE USE OF FOUR DIMENSIONS IN THOUGHT

From this square, let it be supposed that that direc-
tion in which the figures are represented runs to the
left hand. Thus we have a cube (1) running from the
square above, in which the square itself is hidden, but
the letters a, E, I, o, of the conclusion are seen. In this
cube we have the minor premiss and the conclusion in all
their moods, and all the figures represented. With regard
to the major premiss, since the face (2) belongs to the first
wall from the left in the original arrangement, and in this

(2)

—

1
E

•r

(D -

E
A

4 3 2 1

Fiar.

arrangement was characterised by the major premiss in the
mood A, we may say that the wThole of the cube we now
have put up represents the mood A of the major premiss.

Hence the small cube at the bottom to the right in 1,
nearest to the spectator, is major premiss, mood A ; minor
premiss, mood A; conclusion, mood A; and figure the first.
The cube next to it, running to the left, is major premiss,
mood A; minor premiss, mood A; conclusion, mood A;
figure 2.

So in this cube we have the representations of all the
combinations which can occur when the major premiss,
remaining in the mood a, the minor premiss, the conclu-
sion, and the figures pass through their varieties.

In this case there is no room in space for a natural
representation of the moods of the major premiss. To
represent them we must suppose as before thai there is a
fourth dimension, and Btaiting from this cube as base in
1 he Court h direction in four equal Btages, all t he firsl volume

corresponds to major premiss \, the second to major

100

TIIK FOURTH

premiss, mood E, the nexl t<

to mood o.

The cube wo see is as it wore TT^a>"|y wn-y «ag5ios<
which the four-dimensional figure rests. Its section at
any stage is a cube. But a transition in this direction
being transverse to the whole of our space is represented
by no space motion. We can exhibit successive stages of
the result of transference of the cube in that direction,
but cannot exhibit the product of a transference, however
small, in that direction.

To return to the original method of representing our
variables, consider fig. 5Q. These four cubes represent
four sections of the figure derived from the first of them

Fig. 56.

by moving it in the fourth dimension. The first por-
tion of the motion, which begins with 1, traces out a
more than solid body, which is all in the first figure.
The beginning of this body is shown in 1. The next
portion of the motion traces out a more than solid body,
all of which is in the second figure ; the beginning of
this body is shown in 2 ; 3 and 4 follow on in like
manner. Here, then, in one four-dimensional figure we
have all the combinations of the four variables, major
premiss, minor premiss, figure, conclusion, represented,
each variable going through its four varieties. The dis-
connected cubes drawn are our representation in space by
means of disconnected sections of this higher body.

THE USE OF FOUR DIMENSIONS IN THOUGHT lUl

Now it is only a limited number of conclusions which
are true — their truth depends on the particular combina-
tions of the premisses and figures which they accompany.
The total figure thus represented may be called the
universe of thought in respect to these four constituents,
and out of the universe of possibly existing combinations
it is the province of logic to select those which corre-
spond to the results of our reasoning faculties.

We can go over each of the premisses in each of the
moods, and find out what conclusion logically follows.
But this is done in the works on logic ; most simply and
clearly I believe in " Jevon's Logic." As we are only con-
cerned with a formal presentation of the results we will
make use of the mnemonic lines printed below, in which
the words enclosed in brackets refer to the figures, and
are not significative : —

Barbara celarent Darii ferioque [prions].
Caesare Camestris Festino Baroko [secundae].
[Tertia] darapti disamis datisi felapton.
Bokardo ferisson habet [Quarta insuper addit].
Bramantip camenes dimaris ferapton fresison.

In these lines each significative word has three vowels,
the first vowel refers to the major premiss, and gives the
mood of that premiss, "a" signifying, for instance, that
the major mood is in mood a. The second vowel refers
to the minor premiss, and gives its mood. The third
vowel refers to the conclusion, and gives its mood. Thus
(prion's) — of the first figure — the first mnemonic word is
"barbara," and this gives major premiss, mood a; minor
premiss, moo.l .\ ; conclusion, mood a. Accordingly in the
first of our four cubes we mark the lowest left-hand front
cube. To take anot her instance in the third figure " Tori in,"
the word " ferisson " give- us major premiss mood E — <•.(/.,
no m la p, minor premiss mood i ; some m is 8, conclusion,
mood Oj some 8 L8 not P. The region to be marked then

102

THE i 01 LITE MMEXStOH

in the third representative cube is the one in the second
wall to the right for the major premiss, the third wall
from the front for the minor premiss, and the top layer
for the conclusion.

It is easily seen that in the diagram this cube is
marked, and so with all the valid conclusions. The
regions marked in the total region show which com-
binations of the four variables, major premiss, minor
premiss, figure, and conclusion exist.

That is to say, we objectify all possible conclusions, and
build up an ideal manifold, containing all possible com-
binations of them with the premisses, and then out of
this we eliminate all that do not satisfy the laws of logic.
The residue is the syllogism, considered as a canon of
reasoning.

Looking at the shape which represents the totality
of the valid conclusions, it does not present any obvious
symmetry, or easily characterisable nature. A striking
configuration, however, is obtained, if we project the four-
dimensional figure obtained into a three-dimensional one ;
that is, if we take in the base cube all those cubes which
have a marked space anywhere in the series of four
regions which start from that cube.

This corresponds to making abstraction of the figures,
giving all the conclusions which are valid whatever the
figure may be.

Proceeding in this way we obtain the arrangement of
marked cubes shown in fig. 57. We see
that the valid conclusions arc arranged
almost symmetrically round one cube — the
one on the top of the column starting from
AAA. There is one breach of continuity
however in this scheme. One cube is
unmarked, which if marked would give
It is the one which would be denoted by the

THE USE OF FOUR DIMENSIONS IN THOUGHT

103

letters i, E, o, in the third wall to the right, the second
wall away, the topmost layer. Now this combination of
premisses in the mood IE, with a conclusion in the mood
0, is not noticed in any book on logic with which I am
familiar. Let us look at it for ourselves, as it seems
that there must be something curious in connection with
this break of continuity in the poiograpb.

3rd figure!

Pig. 58.

2nd figure.

4th figure.

The propositions i, E, in the various figures are the
following, as Bhown in the accompanying scheme, fig. .58 : —
Kir-f figure : -<>me M is P ; no 8 is M. Second figure :
sonic p is m ; no s is m. Third figure: some M is p ; no
m [g s. Fourth figure : some p is m ; no m is s.

Examining these figures, we see, taking the first, thai
ome M is r and no 8 is M, we have; no conclusion of

104 THE FOURTH DIMENSION

the form S is P in the various moods. It is quite inde-
terminate how the circle representing 8 lies with regard
to the circle representing P. It may lie inside, outside,
or partly inside P. The same is true in the other tigures
2 and o. But when we come to the fourth figure, since
M and S lie completely outside each other, there cannot
lie inside s that part of r which lies inside M. Tsow
we know by the major premiss that some of P does lie
in M. Hence s cannot contain the whole of p. In
words, some r is m, no m is s, therefore s does not contain
the whole of P. If we take P as the subject, this gives
us a conclusion in the mood o about p. Some p is not s.
But it does not give us conclusion about 8 in any one
of the four forms recognised in the syllogism and called
its moods. Hence the breach of the continuity in the
poiograph has enabled us to detect a lack of complete-
ness in the relations which are considered in the syllogism.

To take an instance : — Some Americans (p) are of
African stock (m) ; !No Aryans (s) are of African stock
(m) ; Aryans (s) do not include all of Americans (p).

In order to draw a conclusion about s we have to admit
the statement, " s does not contain the whole of P," as
a valid logical form — it is a statement about s which can
be made. The logic which gives us the form, " some P
is not s," and which does not allow us to give the exactly
equivalent and equally primary form, " s does not con-
tain the Whole of p," is artificial.

And I wish to point out that this artificiality leads
to an error.

If one trusted to the mnemonic lines given above, one
would conclude that no logical conclusion about S can
be drawn from the statement, u some p are M, no M are s."

But a conclusion can be drawn : s does not contain
the whole of P.

It is not that the result is given expressed in another

THE USE OF FOUR DIMEKSlONS IN THOUGHT lUO

form. The mnemonic lines deny that any conclusion
can be drawn from premisses in the moods I, E, respectively

Thus a simple four-dimensional poiograph has enabled
us to detect a mistake in the mnemonic lines which have
been handed down unchallenged from mediaeval times.
To discuss the subject of these lines more fully a logician
defending them would probably say that a particular
statement cannot be a major premiss ; and so deny the
existence of the fourth figure in the combination of moods.

To take our instance : some Americans are of African
stock ; no Aryans are of African stock. He would say
that the conclusion is some Americans are not Aryans ;
and that the second statement is the major. He would
refuse to say anything about Aryans, condemning us to
an eternal silence about them, as far as these premisses
are concerned ! But, if there is a statement involving
the relation of two classes, it must be expressible as a
statement about either of them.

To bar the conclusion, "Aryans do not include the
whole of Americans," is purely a makeshift in favour of
a false classification.

And the argument drawn from the universality of the
major premiss cannot be consistently maintained. It
would preclude such combinations as major o, minor A,
conclusion o — i.e., such as some mountains (m) are not
permanent (p) j all mountains (m) are scenery (s) ; some
scenery (s) is not permanent (r).

This is allowed in " Jevon's Logic," and his omission to
discuss i, E, o, in (he fourth figure, is inexplicable. A
satisfactory poiograph of the Logical Bcheme can be made
admitting the use of the words some, none, or all,
about the predicate a- well as about the subject. Then
we can express the statement, " Aryans do not include the
whole of Americans/' clumsily, but, when its obscurity
i- fathomed, correctly, as "Some Aryans are not all

106* THE FOURTH DIMENSION

Americans." And this method is what is called (lie
"quantification of the predicate."

The laws of formal logic are coincident with the con-
clusions which can be drawn about regions of space, which
overlap one another in the various possible ways. It is
not difficult so to state the relations or to obtain a
symmetrical poiograph. But to enter into this branch of
geometry is beside our present purpose, which is to show
the application of the poiograph in a finite and limited
region, without any of those complexities which attend its
use in regard to natural objects.

If we take the latter — plants, for instance — and, without
assuming fixed directions in space as representative of
definite variations, arrange the representative points in
such a manner as to correspond to the similarities of the
objects, we obtain configuration of singular interest ; and
perhaps in this way, in the making of shapes of shapes,
bodies with bodies omitted, some insight into the structure
of the species and genera might be obtained.

CHAPTER IX

APPLICATION TO KANT'S THEORY OF
EXPERIENCE

When we observe the heavenly bodies we become aware
that they all participate in one universal motion — a
diurnal revolution round the polar axis.

In the case of fixed stars this is most unqualifiedly true,
but in the case of the sun, and the planets also, the single
motion of revolution can be discerned, modified, and
slightly altered by other and secondary motions.

Hence the universal characteristic of the celestial bodies
is that they move in a diurnal circle.

But we know that this one great fact which is true of
them all has in reality nothing to do with them. The
diurnal revolution which they visibly perform is the result
of the condition of the observer. It is because the
observer is on a rotating earth that a universal statement
can be made about all the celestial bodies.

The universal Btatement which is valid about every one
of the celestial bodies is that which docs not concern
them ;ii all, and is but ;i statement of the condition of
t lie observer.

Now there are universal statements of other kinds
which wo can make. We can say thai all objects of
experience are in space and Bubject to the laws o'
geometry.

107

108 THE FOURTH DIMENSION

Dues this mean thai space and all thai if means is due
to a condition of the observer ?

If a universal law in one ease means nothing affecting
the objects themselves, but only a condition of observa-
tion, is this true in every case? There is shown us in
astronomy a vera causa for the assertion of a universal.
Is the same cause to be traced everywhere?

Such is a first approximation to the doctrine of Kant's
critique.

It is the apprehension of a relation into which, on the
one side and the other, perfectly definite constituents
enter — the human observer and the stars — and a trans-
ference of this relation to a region in which the con-
stituents on either side are perfectly unknown.

If spatiality is due to a condition of the observer, the
observer cannot be this bodily self of ours — the body, like
the objects around it, are equally in space.

This conception Kant applied, not only to the intuitions
of sense, but to the concepts of reason — wherever a universal
statement is made there is afforded him an opportunity
for the application of his principle. He constructed a
system in which one hardly knows which the most to
admire, the architectonic skill, or the reticence with regard
to things in themselves, and the observer in himself.

His system can be compared to a garden, somewhat
formal perhaps, but with the charm of a quality more
than intellectual, a besonnenheit, an exquisite moderation
over all. And from the ground he so carefully prepared
with that buried in obscurity, which it is fitting should
be obscure, science blossoms and the tree of real knowledge
grows.

The critique is a storehouse of ideas of profound interest.
The one of which I have given a partial statement leads,
as we shall see on studying it in detail, to a theory of
mathematics suggestive of enquiries in many directions.

APPLICATION TO KANT'S THEORY OF EXPERIENCE 109

The justification for my treatment will be found
amongst other passages in that part'of the transcendental
analytic, in which Kant speaks of objects of experience
subject to the forms of sensibility, not subject to the
concepts of reason.

Kant asserts that whenever we think we think of
objects in space and time, but he denies that the space
and time exist as independent entities. He goes about
to explain them, and their universality, not by assuming
them, as most other philosophers do, but by postulating
their absence. How then does it come to pass that the
world is in space and time to us ?

Kant takes the same position with regard to what we
call nature — a great system subject to law and order.
" How do you explain the law and order in nature ? " we
ask the philosophers. All except Kant reply by assuming
law and order somewhere, and then showing how we can
recognise it.

In explaining our notions, philosophers from other than
the Kantian standpoint, assume the notions as existing
outside us, and then it is no difficult task to show how
they come to us, either by inspiration or by observation.

We ask 4V Why do we have an idea of law in nature ?"
" Because natural processes go according to law," we are
answered, " and experience inherited or acquired, gives us
this notion."

But when we speak about the law in nature we are
speaking about a notion of our own. So all that these
expositors do is to explain our notion by an assumption
of it.

Kant is very different. He supposes nothing. An ex-
perience BUch as ours is Very different from experience
in the abstract. Imagine just simply experience, suc-
cession of states, of consciousness! Why, there would
be no connecting any two together, there would be no

110 THE FOURTH DIMENSION

personal identity, no memory. It is out of a general
experience such as this, which, in respect to anything we
call real, is less than a dream, that Kant shows the
genesis of an experience such as ours.

Kant takes up the problem of the explanation of spar.'.
time, order, and so quite logically does not presuppose
them.

But howT, when every act of thought is of things in
space, and time, and ordered, shall we represent to our-
selves that perfectly indefinite somewhat which is Kant's
necessary hypothesis — that which is not in space or time
and is not ordered. That is our problem, to represent
that which Kant assumes not subject to any of our forms
of thought, and then show some function which working
on that makes it into a " nature " subject to law and
order, in space and time. Such a function Kant calls the
''Unity of Apperception" ; i.e., that which makes our state
of consciousness capable of being woven into a system
with a self, an outer world, memory, law, cause, and order.

The difficulty that meets us in discussing Kant's
hypothesis is that everything we think of is in space
and time — how then shall we represent in space an exis-
tence not in space, and in time an existence not in time?
This difficulty is still more evident when we come to
construct a poiograph, for a poiograph is essentially a
space structure. But because more evident the difficulty
is nearer a solution. If we always think in space, i.e.
using space concepts, the first condition requisite for
adapting them to the representation of non-spatial exis-
tence, is to be aware of the limitation of our thought,
and so be able to take the proper steps to overcome it.
The problem before us, then, is to represent in space an
existence not in space.

The solution is an easy one. It is provided by the
conception of altei nativity.

APPLICATION TO KAKT's THEORY OF EXPERIENCE 111

To get our ideas clear let us go right hack behind the
distinctions of an inner and an outer world. Both of
these, Kant says, are products. Let us take merely states
of consciousness, and not ask the question whether they are
produced or superinduced — to ask such a question is to
have got too far on, to have assumed something of which
we have not traced the origin. Of these states let us
simply say that they occur. Let us now use the word
a " posit " for a phase of consciousness reduced to its
last possible stage of evanescence ; let a posit be that
phase of consciousness of which all that can be said is
that it occurs.

Let a, 6, c, be three such posits. We cannot represent
them in space without placing them in a certain order,
as a, b, c. But Kant distinguishes between the forms
of sensibility and the concepts of reason. A dream in
which everything happens at haphazard would be an
experience subject to the form of sensibility and only
partially subject to the concepts of reason. It is par-
tially subject to the concepts of reason because, although
there is no order of sequence, still at any given time
there is order. Perception of a thing as in space is a
form of sensibility, the perception of an order is a concept
of reason.

We in ust, therefore, in order to get at that process
which Kant supposes to be constitutive of an ordered
experience imagine the posits as in space without
older.

A- we know them they must be in some order, abc,
l>ra, cab, acb, cba, bac, one or another.

To represent them as having no order conceive all
these differenl orders as equally existing. Introduce the
conception of alternat ivit y — let us suppose that the order
abc, and bac, for example, exisl equally, so that we
cannot say about a that it comes before or after 6. This

112 THE FOURTH DIMENSION

would correspond to «i sudden and arbitrary change of a
into 6 and b into a, so that, to use Kant's words, it would
be possible to call one thing by one name at one time
and at another time by another name.

In an experience of this kind we have a kind of chaos,
in which no order exists; it is a manifold not subject to
the concepts of reason.

Now is there any process by which order can be intro-
duced into such a manifold — is there any function of
consciousness in virtue of which an ordered experience
could arise ?

In the precise condition in which the posits are, as
described above, it does not seem to be possible. But
if we imagine a duality to exist in the manifold, a
function of consciousness can be easily discovered which
will produce order out of no order.

Let us imagine each posit, then, as having, a dual aspect.
Let a be la in which the dual aspect is represented by the
combination of symbols. And similarly let b be 26,
c be 3c, in which 2 and b represent the dual aspects
of 6, 3 and c those of c.

Since a can arbitrarily change into b, or into c, and
so on, the particular combinations written above cannot
be kept. We have to assume the equally possible occur-
rence of form such as 2a, 2b, and so on ; and in order
to get a representation of all those combinations out of
which any set is alternatively possible, we must take
every aspect with every aspect. We must, that is, have
every letter with every number.

Let us now apply the method of space represention.

Note. — At the beginning of the next chapter the same
structures as those which follow are exhibited in
more detail and a reference to them will remove
any obscurity wThich may be found in the imme-
diately following passages. They are there carried

APPLICATION TO KANt's THEORY OF EXPERIENCE 113

\> i <y

Fig. 59.

on to a greater multiplicity of dimensions, and the
significance of the process here briefly explained
becomes more apparent.
Take three mutually rectangular axes in space 1, 2, 3
(fig. 59), and on each mark three points,
the common meeting point being the
first on each axis. Then by means of
these three points on each axis we
define 27 positions, 27 points in a
cubical cluster, shown in fig. 60, the
same method of co-ordination being
used as has been described before.
Each of these positions can be named by means of the
axes and the points combined.

Thus, for instance, the one marked by an asterisk can
be called lc, 26, 3c, because it is
opposite to c on 1, to b on 2, to
c on 3.

Let us now treat of the states of
consciousness corresponding to these
positions. Each point represents a
composite of posits, and the mani-
fold of consciousness corresponding
to them is of a certain complexity.
Suppose now the constituents, the points on the axes,
to interchange arbitrarily, any one to become any other,
and also the axes 1, 2, and 3, to interchange amongst
themselves, any one to become any other, and to be sub-
ject to no Bjstem or law. that is to say, that order does
not exist, and that the points which run abc on each axis
may run hoc, and so on.

Then any one of the states of consciousness represented
by the points in the cluster can become any other. We
have a representation of a random consciousness of a
certain degree of complexity,

Fig. 60.

1 1 1

THE FOURTH DIMKNslON

Ic2a3c

Now let US examine carefully one particular case of
arbitrary interchange of the points, (/. A, c\ as one such
case, carefully considered, makes the whole clear.

Consider the points named in the figure lc, 2a, 3c ;

lc, 2c, oa ; la, 2c, 3c, and
examine the effect on them
when a change of order takes
place. Let us suppose, for
instance, that a changes into 6,
and let us call the two sets of
points we get, the one before
and the one after, their change
conjugates.

la2c3c

Fig. 61.

Before the change lc la 'Sc lc 2c 3a la 2c Bo \ r, .

After the change lc 2b 3c lc 2c U lb 2c 3c / ' °

The points surrounded by rings represent the conjugate
points.

It is evident that as consciousness, represented first by
the first set of points and afterwards by the second set of
points, would have nothing in common in its two phases.
It would not be capable of giving an account of itself.
There would be no identity.

If, however, we can find any set of points in the
cubical cluster, which, when any arbitrary change takes
place in the points on the axes, or in the axes themselves,
repeats itself, is reproduced, then a consciousness repre-
sented by those points would have a permanence. It
would have a principle of identity. Despite the no law,
the no order, of the ultimate constituents, it would have
an order, it would form a system, the condition of a
personal identity would be fulfilled.

The question comes to this, then. Can we find a
system of points which is self-conjugate which is such
that when any posit on the axes becomes any other, or

APPLICATION TO KANT's THEORY OF EXPERIENCE 115

lb2d3C

a£~b3C

\C2d~b

IC2

ld2C3t)

Fie. G2.

Self-

whcn any axis becomes any other, such a set is trans-
formed into itself, its identity
is not submerged, but rises
superior to the chaos of its
constituents ?

Such a set can be found.
Consider the set represented
in fig. 62, and written down in
the first of the two lines —

f la 2b 3c lb 2a 3c Ic 2a 3b \c2b3a lb 2c 3a la 2c 3b
conjugate.! lr? 2b 3a lb 2c 3a la 2c 3b la 2b 3c lb 2a 3c lc 2a 3b

If now a change into c and c into a, we get the set in
the second line, which has the same members as are in the
upper line. Looking at the diagram we see that it would
correspond simply to the turning of the figures as a
whole.* Any arbitrary change of the points on the axes,
or of the axes themselves, reproduces the same set.

Thus, a function, by which a random, an unordered, con-
sciousness could give an ordered and systematic one, can
be represented. It is noteworthy that it is a system of
selection. If out of all the alternative forms that only is
attended to which is self-conjugate, an ordered conscious-
ness is formed. A selection gives a feature of permanence.

Can we say that the permanent consciousness is this
selection ?

An analogy between Kant and Darwin comes into light.
That which is swings clear of the fleeting, in virtue of its
presenting a feature of permanence. There is no need
to suppose any function of "attending to." A con-
sciousness capable of giving an account of itself is one
which is characterised by this combination. All com-
binations exist — of this kind is the consciousness which
can give an account of itself. And the very duality which

These figures are described more tally, and extended, in the next
chapter.

110 TllK FOUKTU DtliBNStOK

we have presupposed may be regarded as originated by

a process of selection.

Darwin set himself to explain the origin of the fauna
and flora of the world, lie denied specific tendencies.
He assumed an indefinite variability —that is, chance —
but a chance confined within narrow limits as regards the
magnitude of any consecutive variations. He showed that
organisms possessing features of permanence, if they
occurred would be preserved. So his account of any
structure or organised being was that it possessed features
of permanence.

Kant, undertaking not the explanation of any particular
phenomena but of that which we call nature as a whole,
had an origin of species of his own, an account of the
flora and fauna of consciousness. He denied any specific
tendency of the elements of consciousness, but taking our
own consciousness, pointed out that in which it resembled
any consciousness which could survive, which could give
an account of itself.

He assumes a chance or random world, and as great
and small were not to him any given notions of which he
could make use, he did not limit the chance, the random-
ness, in any way. But any consciousness which is per-
manent must possess certain features — those attributes
namely which give it permanence. Any consciousness
like our own is simply a consciousness which possesses
those attributes. The main thing is that which he calls
the unity of apperception, which we have seen above is
simply the statement that a particular set of phases of
consciousness on the basis of complete randomness will be
self-conjugate, and so permanent.

As with DarwTin so with Kant, the reason for existence
of any feature comes to this — show that it tends to the
I ermanence of that which possesses it.

We can thus regard Kant as the creator of the first of

APPLICATION TO KANT S THEORY OF EXPERIENCE 117

the modem evolution theories. And, as is so often the
case, the first effort was the most stupendous in its scope.
Kant does not investigate the origin of any special part
of the world, such as its organisms, its chemical elements,
its social communities of men. He simply investigates
the origin of the whole — of all that is included in con-
sciousness, the origin of that "thought thing" whose
progressive realisation is the knowahle universe.

This point of view is very different from the ordinary
one, in which a man is supposed to be placed in a world
like that which he has come to think of it, and then to
learn what he has found out from this model which he
himself has placed on the scene.

We all know that there are a number of questions in
attempting an answer to which such an assumption is not
allowable.

Mill, for instance, explains our notion of " law " by an
invariable sequence in nature. But what we call nature
is something given in thought. So he explains a thought
of law and order by a thought of an invariable sequence.
He leaves the problem where he found it.

Kant's theory is not unique and alone. It is one of
B number of evolution theories. A notion of its import
and significance can be obtained by a comparison of it
with other theories.

Thus in Darwin's theoretical world of natural selection
B certain assumption is made, the assumption of indefinite
variability — slight variability it is true, over any appre-
ciable lapse of time, but indefinite in the postulated
epochs of transformation— and a whole chain of results
Is shown f" follow.

This element of chance variation is not, however, an
alt i mate resting place It is a preliminary stage. This
supposing the all is a preliminary step towards finding
nut what is. If everv Kind of ( rganism can come into

118 THE FOURTH DIMENSION

being, those that do survive will present such and such
characteristics. This is the necessary beginning for ascer-
taining what kinds of organisms do come into existence.
And so Kant's hypothesis of a random consciousness is
the necessary beginning for the rational investigation
of consciousness as it is. His assumption supplies, as
it were, the space in which we can observe the pheno-
mena. It gives the general laws constitutive of any
experience. If, on the assumption of absolute random-
ness in the constituents, such and such would be
characteristic of the experience, then, whatever the con-
stituents, these characteristics must be universally valid.

We will nowr proceed to examine more carefully the
poiograph, constructed for the purpose of exhibiting an
illustration of Kant's unity of apperception.

In order to show the derivation order out of non-order
it has been necessary to assume a principle of duality —
we have had the axes and the posits on the axes — there
are two sets of elements, each non-ordered, and it is in
the reciprocal relation of them that the order, the definite
system, originates.

Is there anything in our experience of the nature of a
duality ?

There certainly are objects in our experience wThich
have order and those which are incapable of order. The
two roots of a quadratic equation have no order. No one
can tell which comes first. If a body rises vertically and
then goes at right angles to its former course, no one can
assign any priority to the direction of the north or to the
east. There is no priority in directions of turning. We
associate turnings with no order progressions in a line
with order. But in the axes and points we have assumed
above there is no such distinction. It is the same, whether
we assume an order among the turnings, and no order
among the points on the axes, or, vice versa, an order in

APPLICATION TO KANT'S THEORY OF EXPERIENCE 119

the points and no order in the turnings. A being with
an infinite number of axes mutually at right angles,
with a definite sequence between them and no sequence
between the points on the axes, would be in a condition
formally indistinguishable from that of a creature who,
according to an assumption more natural to us, had on
each axis an infinite number of ordered points and no
order of priority amongst the axes. A being in such
a constituted world would not be able to tell which
was turning and which was length along an axis, in
order to distinguish between them. Thus to take a per-
tinent illustration, we may be in a world of an infinite
number of dimensions, with three arbitrary points on
each — three points whose order is indifferent, or in a
world of three axes of arbitary sequence with an infinite
number of ordered points on each. We can't tell which
is which, to distinguish it from the other.

Thus it appears the mode of illustration which we
have used is not an artificial one. There really exists
in nature a duality of the kind which is necessary to
explain the origin of order out of no order — the duality,
namely, of dimension and position. Let us use the term
group for thai system of points which remains unchanged,
whatever arbitrary change of its constituents takes place.
We not ice that a group involves a duality, is inconceivable
w it liout a dual it v.

Thus, according to Kant, the primary element of ex-
perience is the group, and the theory of groups would be
the mosl fundamental branch of science. Owing to an
expression in the critique the authority of Kant is some-
times adduced against the assumption of more than three
dimensions to space. It seems to me, however, thai the
whole tendency of his t heory li*'< in t he opposite direct ion,
and points to a perfect duality between dimension and
position in a dimension,

120 THE FOURTH DIMENSION

Tf the order and the law we see is due to the conditions
of conscious experience, we must conceive nature as
spontaneous, free, subject to no predication that we can
devise, but, however apprehended, subject to our logic.

And our logic is simply spatiality in the general sense
— that resultant of a selection of the permanent from the
impermanent, the ordered from the unordered, by the
means of the group and its underlying duality.

We can predicate nothing about nature, only about the
way in which we can apprehend nature. All that we can
say is that all that which experience gives us will be con-
ditioned as spatial, subject to our logic. Thus, in exploring
the facts of geometry from the simplest logical relations
to the properties of space of any number of dimensions,
we are merely observing ourselves, becoming aware of
the conditions under which we must perceive. Do any
phenomena present themselves incapable of explanation
under the assumption of the space we are dealing with,
then we must habituate ourselves to the conception of a
higher space, in order that our logic may be equal to the
task before us.

We gain a repetition of the thought that came before,
experimentally suggested. If the laws of the intellectual
comprehension of nature are those derived from con-
sidering her as absolute chance, subject to no law save
that derived from a process of selection, then, perhaps, the
order of nature requires different faculties from the in-
tellectual to apprehend it. The source and origin of
ideas may have to be sought elsewhere than in reasoning.

The total outcome of the critique is to leave the
ordinary man just where he is, justified in his practical
attitude towards nature, liberated from the fetters of his
own mental representations.

The truth of a picture lies in its total effect. It is vain
to seek information about the landscape from an examina-

APPLICATION TO KAKT'S THEORY OF EXPERIENCE 121

tion of the pigments. And in any method of thought it
is the complexity of the whole that brings us to a know-
ledge of nature. Dimensions are artificial enough, but in
the multiplicity of them we catch some breath of nature.

We must therefore, and this seems to me the practical
conclusion of the whole matter, proceed to form means of
intellectual apprehension of a greater and greater degree
of complexity, both dirnensionally and in extent in any
dimension. Such means of representation must always
be artificial, but in the multiplicity of the elements with
which we deal, however incipiently arbitrary, lies our
chance of apprehending nature.

And as a concluding chapter to this part of the book,
I will extend the figures, which have been used to repre-
sents Kant's theory, two steps, so that the reader may
have the opportunity of looking at a four-dimensional
figure which can be delineated without any of the special
apparatus, to the consideration of which I shall subse-
quently pass on.

CHAPTER X

A FOUR-DIMENSIONAL FIGURE

The method used in the preceding chapter to illustrate
the problem of Kant's critique, gives a singularly easy
and direct mode of constructing a series of important
figures in any number of dimensions.

We have seen that to represent our space a plane being
must give up one of his axes, and similarly to represent
the higher shapes we must give up one amongst our
three axes.

But there is another kind of giving up which reduces
the construction of higher shapes to a matter of the
utmost simplicity.

Ordinarily we have on a straight line any number of
positions. The wealth of space in position is illimitable,
while there are only three dimensions.

I propose to give up this wealth of positions, and to
consider the figures obtained by taking just as many
positions as dimensions.

In this way I consider dimensions and positions as two
" kinds," and applying the simple rule of selecting every
one of one kind with every other of every other kind,
get a series of figures which are noteworthy because
they exactly fill space of any number of dimensions
(as the hexagon fills a plane) by equal repetitions of
themselves,

123

A FOUR-DIMENSIONAL FIGURE 123

The rule will be made more evident by a simple
application.

Let us consider one dimension and one position. I will
call the axis i, and the position o.

Here the figure is the position o on the line i. Take
now two dimensions and two positions on each.

We have the two positions o ; 1 on i, and the two
positions o, 1 on j, fig. 63. These give

1 J rise to a certain complexity. I will

let the two lines i and j meet in the

position I call o on each, and I will

consider i as a direction starting equally

£' • from every position on j, and j as

starting equally from every position on i. We thus

obtain the following figure : — A is both oi and oj, B is 1 i

A C and oj, and so on as shown in fig. 63fr.

<^;ij The positions on AC are all oi positions.

They are, if wTe like to consider it in

that way, points at no distance in the i

direction from the line AC. Wo can

call the line AC the oi line. Similarly

the points on AB are those no distance

Fig. G3&. from ah in the^' direction, and we can

call them oj points and the line A\\ the oj line. Again,

the line CD can be called the 1/ line because the points

on it are at a distance, 1 in the;' direction.

We have then four positions or points named as shown,
and. considering directions and positions as " kinds," we
have tin- combination of two kinds with two kinds. >.'<>w\
selecting every one of one kind with every other of every
other kind will mean that we take 1 of the kind /and

124

THE FOUKTH DIMENSION

with il o of the kind j ; and t hen, that we take o of the
kind i and with it 1 of the kind j.
Thus we gel a pair of positions lying in the straight
C line lie, fig. 04. We can call this pair 10
and 01 if we adopt the plan of mentally,
adding an i to the first and a j to the
second of the symbols written thus — 01
is a short expression for Oi, \j.
tig. G4. Coming now to our space, we have three

dimensions, so we take three positions on each. These
positions I will suppose to be at equal distances along each

Fig. G5.

axis. The three axes and the three positions on each are
shown in the accompanying diagrams, fig. 65, of which
the first represents a cube with the front faces visible, the
second the rear faces of the same cube ; the positions I
will call 0, 1, 2 ; the axes, i,j, k. I take the base abc as
the starting place, from which to determine distances in
the k direction, and hence every point in the base ABC
will be an ok position, and the base abc can be called an
ok plane.

In the same way, measuring the distances from the face
ADC, we see that every position in the face adc is a oi
position, and the whole plane of the face may be called an
oi) plane. Thus we see that with the introduction of a

A FOUtl-DlMENSIONAL FlGUttE

125

new dimension the signification of a compound symbol,
such as u oi" alters. In the plane it meant the line AC.
In space it means the whole plane ACD.

Now, it is evident that we have twenty-seven positions,
each of them named. If the reader will follow this
nomenclature in respect of the positions marked in the
figures he will have no difficulty in assigning names to
each one of the twenty-seven positions. A is oi, oj, ok.
It is at the distance 0 along i, 0 along j, 0 along k, and
io can be written in short 000, where the ijk symbols
are omitted.

The point immediately above is 001, for it is no dis-
tance in the i direction, and a distance of 1 in the h
direction. Again, looking at B, it is at a distance of 2
from A, or from the plane ADC, in the i direction, 0 in the
j direction from the plane abd, and 0 in the k direction,
measured from the plane abc. Hence it is 200 written
for 2i, Oj, 0k

Now, out of these twenty-seven "things " or compounds
of position and dimension, select those which are given by
the rule, every one of one kind with every other of every
other kind.

Take 2 of the i kind. With this

we must have a 1 of the j kind,

and then by the rule we can only

have a 0 of the /.; kind, for if we

had any other of the k kind we

should repeat one of the kinds we

already had. In 2i, \j, \k, for

instance, 1 is repeated. The point

we obtain is that marked 210, fig. (>('».

Proceeding in this way, we pick out the following

cluster of points, fig. 07. They are joined by line-,

(letted where they are hidden by the body of the cube,

and we .-<■<• that they form a figure a hexagon which

Fig. 66.

L26

Till: FOURTH DIMENSION

could be taken out of the cube and placed on a plane

It is a figure which will fill a
plane by equal repetitions of itself.
The plane being representing this
construction in his plane would
take three squares to represent the
cube. Let us suppose that he
takes the ij axes in his space and
k represents the axis running out
of his space, fig. 68. In each of
the three squares shown here as drawn separately he
could select the points given by the rule, and he would

201

Fig. 67.

Fig. 68.

then have to try to discover the figure determined by
the three lines drawn. The line from 210 to 120 is
given in the figure, but the line from 210 to 201 or FG
is not given. He can determine FG by making another
set of drawings and discovering in them what the relation
between these two extremities is.

102^

02.

3 y^

2JO yS

120

Fig.

00.

Let him draw the % and h axes in his plane, fig. G(J
The j axis then runs out and he has the accompanying
figure. In the first of these three squares, fig. G9, he can

A FOUR-DIMENSIONAL FIGURE

to*

J rw .

pick out by the rule the two points 201, 102 — G, and K.
Here they occur in one plane and he can measure the
distance between them. In his first representation they
occur at G and k in separate figures.

Thus the plane being would find that the ends of each
of the lines was distant by the diagonal of a unit square
from the corresponding end of the last and he could then
place the three lines in their right relative position.
Joining them he would have the figure of a hexagon.
We may also notice that the plane being could make
a representation of the whole cube
simultaneously. The three squares,
shown in perspective in fig. 70, all
lie in one plane, and on these the
plane being could pick out any
selection of points just as well as on
three separate squares. He would
obtain a hexagon by joining the
points marked. This hexagon, as
drawn, is of the right shape, but it would not be so if
actual squares were used instead of perspective, because
the relation between the separate squares as they lie in
the plane figure is not their real relation. The figure,
however, as thus constructed, would give him an idea of
the correct figure, and he could determine it accurately
bv remembering that distances in each square were
correct, but in passing from one square to another their
distance in the third dimension had to be taken into
account.

Coming now to the Ggure made by selecting according
to our rule from the whole mass of points given by four
axes and four positions in each, we must first draw a
catalogue figure in which the whole assemblage is shown.

Wo can represent this assemblage of points bv four
Bolid figures. The firs! giving all those positions which

Fig. 70.

128

TI1K FOURTH DIMENSION

are at a distance o from our space in the fourth dimen-
sion, the second showing all those that are at a distance 1,
and so on.

These figures will each be cubes. The first two are
drawn showing the front faces, the second two the rear
faces. We will mark the points 0, 1,2, 3, putting points
at those distances along each of these axes, and suppose

Fig. 71.

all the points thus determined to be contained in solid
models of which our drawings in fig. 71 are represen-
tatives. Here we notice that as on the plane Oi meant
the whole line from which the distances in the i direction
was measured, and as in space Qi means the whole plane
from which distances in the i direction are measured, so
now Oh means the whole space in which the first cube
stands — measuring away from that space by a distance
of one we come to the second cube represented.

A FOUR-DIMENSIONAL FIGURE

129

Now selecting according to the rule every one of one
kind with every other of every other kind, we must take,
for instance, 3i, 2j, Ik, Oh. This point is marked
3210 at the lower star in the figure. It is 3 in the
i direction, 2 in the j direction, 1 in the k direction,
0 in the h direction.

With Si we must also take \j, 2k, Oh. This point
is shown by the second star in the cube Oh.

Fig. 72.

In the first cube, since all the points are Oh point-.
we can only have varieties in which /,./', k, arc accom-
panied by 3, 2, 1.

The points determined arc marked off in the diagram
fig. 7L\ and lines arc drawn joining the adjacent pairs
in each figure, the lines being dotted when they pass
within the substance of the cube in the firsl two diagrams.

Opposite each point, on one Bide or the other of each

130

TIIK Kdl'llTII DIMENSION

cube, is written its name. It will be noticed thai the
figures are symmetrical right and left ; and right and
left the first two numbers are simply interchanged.

Now this being our selection of points, what figure do
they make when all are put together in their proper
relative positions ?

To determine this we must find the distance between
corresponding corners of the separate hexagons.

0312

1032

At 2031 "
0321

2130

1320

Fig. 73.

To do this let us keep the axes i, j, in our space, and
draw h instead of k, letting k run out in the fourth
dimension, fig. 73.

Here we have four cubes again, in the first of which all
the points are Ok points ; that is, points at a distance zero
in the k direction from the space of the three dimensions
ijh. We have all the points selected before, and some
of the distances, which in the last diagram led from figure
to figure are shown here in the same figure, and so capable

a FOUR-nnrrcxsioNAL figtrk

131

of measurement. Take for instance the points 3120 to
3021, which in the first diagram (fig. 72) lie in the first
and second figures. Their actual relation is shown in
fig. 73 in the cube marked 2k, where the points in ques-
tion are marked with a * in fig. 73. We see that the
distance in question is the diagonal of a unit square. In
like manner we find that the distance between corres-
ponding points of any two hexagonal figures is the
diagonal of a unit square. The total figure is now easily
constructed. An idea
of it may be gained by
drawing all the four
cubes in the catalogue
figure in one (fig. 74).
These cubes are exact
repetitions of one
another, so one draw-
ing will serve as a
representation of the
whole series, if we
take care to remember
where we are, whether
in a Oh, a \h, a 2h,
or a 3h figure, when
we pick out the points required. Fig. 74 is a represen-
tation of all the catalogue cubes put in one. For the
Bake of clearness the front faces and the back faces of
(his cube are represented separately.

The figure determined by the selected points is shown
1 elow.

In putting (he sections together some of the outlines
in them disappear. The line tw for instance is not
wanted.

We notice thai PQTW and TWBS are each (he half

of a hexagon. Now QV and vb Lie in one straight line.

Fie. 71.

132

TIIK ForUTII DIMENSION

Hence these two hexagons fit together, forming one
hexagon, and t lie line t\v is only wanted when we con-
sider a section of the whole tigure, we thus obtain the
solid represented in the lower part of fig. 74. Equal
repetitions of this figure, called a tetrakaidecagon, will
fill up three-dimensional space.

To make the corresponding four-dimensional figure we
have to take five axes mutually at right angles with five
points on each. A catalogue of the positions determined
in five-dimensional space can be found thus.

Take a cube with five points on each of its axes, the
fifth point is at a distance of four units of length from the
first on any one of the axes. And since the fourth dimen-
sion also stretches to a distance of four we shall need to

represent the succes-
sive sets of points at
distances 0, 1, 2, 3,4,
in the fourth dimen-
sionSjfive cubes. Now
all of these extend to
no distance at all in
the fifth dimension.
To represent what
lies in the fifth dimen-
sion we shall have to
drawT, starting from
each of our cubes, five
similar cubes to re-
present the four steps
on in the fifth dimension. By this assemblage we get a
catalogue of all the points shown in fig. 75, in which
L represents the fifth dimension.

Now, as wTe saw before, there is nothing to prevent us
from putting all the cubes representing the different
stages in the fourth dimension in one figure, if we take

3LM

1LN

OLM

OH 1H

Fisr. 75.

3H 4H

A FOUK-DIM EN SIGNAL FIG U lit! 133

note when we look at it, whether we consider it as a ()//, a
\h, a 2h, etc., cube. Putting then the Oh, \h, 2h, oh, 4h
cubes of each row in one, we have five cubes with the sides
of each containing five positions, the first of these five
cubes represents the 01 points, and has in it the i points
from 0 to 4, the j points from 0 to 4, the k points from
0 to 4, while we have to specify with regard to any
selection we make from it, whether we regard it as a Oh,
a Ih, a 2h, a oh, or a Ah figure. In fig. 76 each cube
is represented by two drawings, one of the front part, the
other of the rear part.

Let then our five cubes be arranged before us and our
selection be made according to the rule. Take the first
figure in which all points are 01 points. We cannot
have 0 with any other letter. Then, keeping in the first
figure, which is that of the 01 positions, take first of all
that selection which always contains ]h. We suppose,
therefore, that the cube is a \h cube, and in it we take
i,j, k in combination with 4, 3, 2 according to the rule.

The figure we obtain is a hexagon, as shown, the one
in front. The points on the right hand have the same
figures as those on the left, with the first two numerals
interchanged. Next keeping still to the Ol figure let
us suppose that the cube before us represents a section
at a distance of 2 in the h direction. Let all the points
in it be considered as 2h points. We then have a Ol, 2h
region, and have the sets ijk and 431 left over. We
must then pick out in accordance with our rule all such
point - as H, 3j, 1/.'.

These are shown in the figure and we find that we can
draw thcni without confusion, forming the second hexagon
from the front. Going on in this way it will be seen
that in each of the five figures a set of hexagons is picked
out, which put together Form a three-space figure some-
thing like the tetrakaidecagon.

* «

A FOUR-DIMENSIONAL FIGURE 135

These separate figures are the successive stages in
which the whole four-dimeusional figure in which they
cohere can be apprehended.

The first figure and the last are tetrakaidecagons.
These are two of the solid boundaries of the figure. The
other solid boundaries can be traced easily. Some of
them are complete from one face in the figure to the
corresponding face in the next, as for instance the solid
which extends from the hexagonal base of the first figure
to the equal hexagonal base of the second figure. This
kind of boundary is a hexagonal prism. The hexagonal
prism also occurs in another sectional series, as for
instance, in the square at the bottom of the first figure,
the oblong at the base of the second and the square at
the base of the third figure.

Other solid boundaries can be traced through four of
the five sectional figures. Thus taking the hexagon at
the top of the first figure we find in the next a hexagon
also, of which some alternate sides are elongated. The
top of the third figure is also a hexagon with the other
set of alternate rules elongated, and finally we come in
the fourth figure to a regular hexagon.

These four sections are the sections of a tetrakaidecagon
as can be recognised from the sections of this figure
which we have had previously. Hence the boundaries
are of two kinds, hexagonal prisms and tetrakaidecagons.

These four-dimensional figures exactly fill four-dimen-
Bional ppace by equal repetitions of themselves.

CHAPTER XI

NOMENCLATURE AND ANALOGIES PRELIM-
INARY TO THE STUDY OF FOUR-DIMEN-
SIONAL FIGURES

In the following pages a method of designating different
regions of space by a systematic colour scheme has been
adopted. The explanations have been given in such a
manner as to involve no reference to models, the diagrams
will be found sufficient. But to facilitate the study a
description of a set of models is given in an appendix
which the reader can either make for himself or obtain.
If models are used the diagrams in Chapters XI. and XII.
will form a guide sufficient to indicate their use. Cubes
of the colours designated by the diagrams should be picked
out and used to reinforce the diagrams. The reader,

in the following description, should
suppose that a board or wall
stretches away from him, against
which the figures are placed.

Take a square, one of those
shown in Fig. 77 and give it a
neutral colour, let this colour be
called " null," and be such that it
makes no appreciable difference

Fig. 77.

IL'G

KOMEXCLATUHE and analogies

137

to any colour with which it mixed. If there is no
such real colour let us imagine such a colour, and
assign to it the properties of the number zero, which
makes no difference in any number to which it is
added.

Above this square place a red square. Thus we symbolise
the going up by adding red to null.

Away from this null square place a yellow square, and
represent going away by adding yellow to null.

To complete the figure we need a fourth square.
Colour this orange, which is a mixture of red and
yellow, and so appropriately represents a going in a
direction compounded of up and away. We have thus
a colour scheme which will serve to name the set of
squares drawn. We have two axes of colours — red and

yellow— and they may oc-
cupy as in the figure the
direction up and away, or
they may be turned about ;
in any case they enable us
to name the four squares
drawn in their relation to
one another.

Now take, in Fig. 78,
nine squares, and suppose
that at the end of the
going in any direction the
colour started with repeats itself.
We obtain a square named as shown.
Lei us dow, in fig. 79, Buppose the number of squares to
be increased, keeping siill to the principle of colouring
already used.

Here the nulls remain four in number. There
are three reds between the firsl null and the null
above it, three yellows between the first null and the

"7"

"T"

~~1#
&»

«*

"7"

r ■_-. 78.

138

tin; fourth dimension

null beyond it, while the oranges increase in a double
way.

«>

4»

Fig. 79.

Red

Null

Orange

Yellow
Fig. 80.

Red
Null

Suppose this process of enlarging the number of the
Nnll Yellow Null squares to be indefinitely pursued and
the total figure obtained to be reduced
in size, we should obtain a square of
which the interior wras all orange,
while the lines round it were red and
yellow, and merely the points null
colour, as in fig. 80. Thus all the points, lines, and the
area would have a colour.

We can consider this scheme to originate thus : — Let
a null point move in a yellow direction and trace out a
yellow line and end in a null point. Then let the whole
line thus traced move in a red direction. The null points
at the ends of the line will produce red lines, and end in

NOMENCLATURE AND ANALOGIES

139

F\z. 81,

null points. The yellow line will trace out a yellow and
red, or orange square.

Now, turning back to fig. 78, we see that these two
ways of naming, the one we started with and the one we
arrived at, can be combined.

By its position in the group of four squares, in fig. 77,
the null square has a relation to the yellow and to the red
directions. We can speak therefore of the red line of the
null square without confusion, meaning thereby the line

ab, fig. 81, which runs up from the
initial null point A in the figure as
drawn. The yellow line of the null
square is its lower horizontal line AC
as it is situated in the figure.

If we wish to denote the upper
yellow line bd, fig. 81, we can speak
of it as the }Tellow r line, meaning
the yellow line which is separated
from the primary yellow line by the red movement.

In a similar way each of the other squares has null
points, red and yellow lines. Although the yellow square
is all yellow, its line CD, for instance, can be referred to as
its red line.

This nomenclature can be extended.
If the eight cubes drawn, in fig. 82, are put close
together, as on the right hand of the diagram, they form
a cube, and in them, as thus arranged, a going up is
represented by adding red to the zero, or null colour, a
going away by adding yellowT, a going to the right by
adding white. White is used as a colour, as a pigment,
which produces a colour change in the pigment 9 with which
it is mixed. From whatever cube of the lower set we
Btart, a motion up brings us to a cube showing a change
t<> red, thus light yellow becomes light yellow red, or
light orange, which i> called ochre. And going to the

140

TlIK Kot'UTH i)lMKNsioS-

right from the null on the left we have a change involving
the introduction of white, while the yellow change runs
from front to back. There are three colour axes — the red,

Jran

yOc\

Red

Pink

ello

iLierht

EL
o"
*

Null

White

.Light
yellow

Fie:. 82.

the white, the yellow — and these run in the position the
cubes occupy in the drawing — up, to the right, away — but
they could be turned about to occupy any positions in space.

Null /White/ Null

'Yellow/ Ljght /Vellov
yellow

«►

•

Third
layer.

Second
layer.

Fisr. 63.

We can conveniently represent a block of cubes by

three sets of squares, representing each the base of a cube.

Thus the block, fig. 83, can be represented by the

NOMENCLATURE AND ANALOGIES

141

layers on the right, Here, as in the case of the plane.

the initial colours repeat themselves at the end of the

series.

Proceeding now to increase the number of the cubes

we obtain fig. 84,
in which the initial
letters of the colours
are given instead of
their full names.

Here we see that
there are four null
cubes as before, but
the series which spring
from the initial corner
will tend to become
lines of cubes, as also
the sets of cubes
parallel to them, start-
ing from other corners.
Thus, from the initial
null springs a line of
red cubes, a line of
white cubes, and a line
of yellow cubes.

If the number of the
cubes is largely in-
creased, and the size
of the whole cube is
diminished, we get
a cube with null
points, and the cd.

/ n-

/ vvh. j

f wh..

/ wh./ n.' /

/ y- • /

l. y. /

I. y. /

1 l/ >'• /

' y- A

y./'l.

y./ I.

y/ >'• /

y. Ay

./i.y

•/i-y

/ y- /

/ n

• / wh./

vvh./

wh. /

n. /

/ r'

/ P- j

f P- /

/ p. / r. /

/ or /

oc. /

oc. / or. /

or. / oc. / oc. / oc. / or. /

or. / oc.

/ oc.

/ or. /

/ r-

/ P- /

p- /

r. /

/ r*

/ p- ,

/ p- /

/ p. / r. /

/ or' /

oc- /

oc. /

oc. / or. /

or. / oc. / oc. / oc. /or. /

or./ oc.

/ oc.

/ or. /

/ r

•/ P- /

p. /

p- /

r. /

/ r.

/ p, ,

/ p, /

/ P,/ r* /

/ or' /

oc. /

oc. / or. /

or. / oc . / oc. / oc. / or. /

/ i

or. / oc.

/ oc.

/ or. /

/ r

/ P. /

f p- /

p-/

r. /

/ n.

/ vvhv

' wh.

/wh,/ ri. /

/ y. /

i-y./

i-y-/

I. y./ y. /

' y. /i.

y-/i.

y*A

y-/ y- /

y / l-y

./ I. y

./]. y.

/ y* /

/x n

. / wh..y

wh . /

wh ./

n. /

Pig. 84.

coloured with those three colours.

The light yellow cubes increase in two ways, forming
ultimately a Bheel of cubes, and the Bame is true <•!
the orange and pink sets. Hence, ultimately the cube

142

THE FOURTH DIMENSION

Null

thus formed would have rod, white, and yellow lines
surrounding pink, orange, and light yellow faces. The
ochre cubes increase in three ways, and hence ulti-
mately the whole interior of the cube would be coloured
ochre.

We have thus a nomenclature for the points, lines,
faces, and solid content of a cube, and it can be named
as exhibited in fig. 85.

We can consider the cube to be produced in the

following way. A null point
moves in a direction to which
we attach the colour indication
yellow ; it generates a yellow line
and ends in a null point. The
yellow line thus generated moves
in a direction to which we give
the colour indication red. This
lies up in the figure. The yellow-
line traces out a yellow, red, or
orange square, and each of its null points trace out a
red line, and ends in a null point.

This orange square moves in a direction to which
we attribute the colour indication white, in this case
the direction is the right. The square traces out a
cube coloured orange, red, or ochre, the red lines trace
out red to white or pink squares, and the yellow
lines trace out light yellow squares, each line ending
in a line of its own colour. While the points each
trace out a null + white, or wThite line to end in a null
point.

Now returning to the first block of eight cubes we can
name each point, line, and square in them by reference to
the colour scheme, which they determine by their relation
to each other.

Thus, in fig. 86, the null cube touches the red cube by

NOMENCLATURE AND ANALOGIES

143

a light yellow square; it touches the yellow cube by a

pink square, and touches

the white cube by an
orange square.

There are three axes
to which the colours red,
yellow, and white are
assigned, the faces of
each cube are designated

range/

S ,

^^Ochre/

S S

\ yellow \

Fiff. 86

by taking these colours in pairs. Taking all the colours
together we get a colour name for the solidity of a cube.
Let us now ask ourselves how the cube,7 could be pre-
sented to the plane being. Without going into the question
of how he could have a real experience of it, let us see
how, if we could turn it about and show it to him, he,
under his limitations, could get information about it.
If the cube were placed with its red and yellow axes
against a plane, that is resting against it by its orange

Null White

g^ed <£& Red

Null White

face previously perceived

Null wh'.

Fig. 87.

face, the plane being would observe a square surrounded
by red and yellow lines, and having null points. Sec the
dotted square, fig. 87.

We could turn the cube about the red line so thai
a different face comes into juxtaposition with the plane.

Suppose the cube turned abotil the red line* A^ [<

144 THE FOURTH DIMENSION

is turning from its first position all of it excepi the red
line leaves the plane — goes absolutely out of tin- range

of the plane being's apprehension. But when the yellow
line points straight out from the plane then the pink
face comes into contact with it. Thus the same red line
remaining as he saw it at first, now towards hi m comes
a face surrounded by white and red lines.

If we call the direction to the right the unknown
direction, then the line he saw before, the yellow line,
goes out into this unknown direction, and the line which
before went into the unknown direction, comes in. It
comes in in the opposite direction to that in which the
yellowr line ran before ; the interior of the face now

against the plane is pink. It is

a property of two lines at right

angles that, if one turns out of

O A a given direction and stands at

right angles to it, then the other

B of the two lines comes in, but

Fio. 8g runs the opposite way in that

given direction, as in fig. 88.

Now these two presentations of the cube would seem,

to the plane creature like perfectly different material

bodies, with only that line in common in which they

both meet.

Again our cube can be turned about the yellow line.
In this case the yellow square would disappear as before,
but a new square would come into the plane after the
cube had rotated by an angle of 90° about this line.
The bottom square of the cube would come in thus
in figure 89. The cube supposed in contact with the
plane is rotated about the lower yellow line and then
the bottom face is in contact with the plane.

Here, as before, the red line going out into the un-
known dimension, the white line which before ran in the

NOMENCLATURE AND ANALOGIES

145

unknown dimension would come in downwards in the
opposite sense to that in which the red line ran before.

Now if we use i, j, h, for the three space directions,
i left to right, j from near away, k from below up ; then,
using the colour names for the axes, we have that first
of all white runs i, yellow runs j, red runs k; then after

rsr appearance

White

White .5*

NuJJy.t Null White NuJI
LYellow

Fig. 89.

the first turning round the k axis, white runs negative,;';
yellow runs i, red runs k ; thus we have the table : —

1-1 position white yellow red

2nd position yellow white — red
8rd position red yellow white —

Here white with a negative sign after it in the column
under / moans that white runs in the negative sense of
the j direction.

We may express the fact in the following way: —
In the plane there is room for two axes while the body
has three. Therefore in the plane we can represent any
two. If we want to keep the axis that goes in the
unknown dimension always running in the positive sense,
then the axis which originally ran in the unknown

I l'i THE FOURTH DIMENSION

dimension (the white axis) must come in in the negative
Bense of thai axis which goes out of the plane into the
unknown dimension.

It is obvious that the unknown direction, the direction
in which the white line runs at first, is quite distinct from
any direction which the plane creature knows. The white
line may come in towards him, or running down. If he
is looking at a square, which is the face of a cube
(looking at it by a line), then any one of the bounding lines
remaining unmoved, another face of the cube may come
in, any one of the faces, namely, which have the white line
in them. And the white line comes sometimes in one
of the space directions he knows, sometimes in another.

Now this turning which leaves a line unchanged is
something quite unlike any turning he knows in the
plane. In the plane a figure turns round a point. The
square can turn round the null point in his plane, and
the red and yellow lines change places, only of course, as
with every rotation of lines at right angles, if red goes
where yellow went, yellow comes in negative of red's old
direction.

This turning, as the plane creature conceives it, we
should call turning about an axis perpendicular to the
plane. What he calls turning about the null point we
call turning about the white line as it stands out from
his plane. There is no such thing as turning about a
point, there is ahvays an axis, and really much more turns
than the plane being is aware of.

Taking now a different point of view, let us suppose the
cubes to be presented to the plane being by being passed
transverse to his plane. Let us suppose the sheet of
matter over which the plane being and all objects in his
world slide, to be of such a nature that objects can pass
through it without breaking it. Let us suppose it to be
of the same nature as the film of a soap bubble, so that

NOMENCLATURE AND ANALOGIES

14'

Null

Nulli .

it closes around objects pushed through it, and, however
the object alters its shape as it passes through it, let us
suppose this film to run up to the contour of the object
in every part, maintaining its plane surface unbroken.

Then we can push a cube or any object through the
film and the plane being who slips about in the film
will know the contour of the cube just and exactly where
the film meets it.

Fig. 90 represents a cube passing through a plane film.

The plane being now comes into
contact with a very thin slice
of the cube somewhere between
the left and right hand faces.
This very thin slice he thinks
of as having no thickness, and
consequently his idea of it is
what we call a section. It is
bounded by him by pink lines
front and back, coming from
the part of the pink face he is
in contact with, and above and belowT, by light yellow
lines. Its corners are not null-coloured points, but white
points, and its interior is ochre, the colour of the interior
of the cube.

If now we suppose the cube to be an inch in each
dimension, and to pass across, from right to left, through
the plane, then we should explain the appearances pre-
sented to the plane being by saying : First of all you
have the face of a cube, this lasts only a moment ; then
you have a figure of the same shape but differently
coloured. This, which appears not to move to you in any
direction which you know of, is really moving transverse
to your plane wold. Its appearance is unaltered, but
each momenl it is something different — a section further
on, in the white, the unknown dimension. Finally, at the

Fig. 90.

U8 THE FOUBTH DIMENSION

end of the mi mite, a face comes in exactly like the lace
you first saw. This finishes up the cube — it is the further
lace in the unknown dimension.

The white line, which extends in length just like the
red or the yellow, you do not see as extensive ; you appre-
hend it simply as an enduring white point. The null
point, under the condition of movement of the cube,
vanishes in a moment, the lasting white point is really
your apprehension of a white line, running in the unknown
dimension. In the same way the red line of the face by
which the cube is first in contact with the plane lasts only
a moment, it is succeeded by the pink line, and this pink
line lasts for the inside of a minute. This lasting pink
line in your apprehension of a surface, which extends in
two dimensions just like the orange surface extends, as you
know it, when the cube is at rest.

But the plane creature might answer, " This orange
object is substance, solid substance, bounded completely
and on every side."

Here, of course, the difficulty comes in. His solid is our
surface — his notion of a solid is our notion of an abstract
surface with no thickness at all.

We should have to explain to him that, from every point
of what he called a solid, a new dimension runs away,
From every point a line can be drawn in a direction
unknown to him, and there is a solidity of a kind greater
than that which he knows. This solidity can only be
realised by him by his supposing an unknown direction,
by motion in which what he conceives to be solid matter
instantly disappears. The higher solid, however, which
extends in this dimension as well as in those which he
knows, lasts when a motion of that kind takes place,
different sections of it come consecutively in the plane of
his apprehension, and take the place of the solid which he
at first conceives to be all. Thus, the higher solid — our

NOMENCLATURE AND ANALOGIES 140

solid in contradistinction to his area solid, his two-
dimensional solid, must be conceived by him as something
which has duration in it, under circumstances in which his
matter disappears out of his world.

We may put the matter thus, using the conception of
motion.

A null point moving in a direction away generates a
yellow line, and the yellow line ends in a null point. We
suppose, that is, a point to move and mark out the
products of this motion in such a manner. Now
suppose this whole line as thus produced to move in
an upward direction ; it traces out the two-dimensional
solid, and the plane being gets an orange square. The
null point moves in a red line and ends in a null point,
the yellow line moves and generates an orange square and
ends in a yellow line, the farther null point generates
a red line and ends in a null point. Thus, by move*
ment in two successive directions known to him, he
can imagine his two-dimensional solid produced with all
its boundaries.

Now we tell him : " This whole two-dimensional solid
can move in a third or unknown dimension to you. The
null point moving in this dimension out of your world
generates a white line and ends in a null point. The
yellow line moving generates a light yellow two-
dimensional solid and ends in a yellow line, and thus
two-dimensional solid, lying end on to your plane world, is
bounded on the far side by the oilier yellow line. In
the same way each of the lines surrounding your square
traces out an area, just like the orange area you know.
Bui there is something new produced, something which
you had do idea of before; it is that which is produced by
the movement of the orange square. That, than which
you can imagine nothing more solid, itself moves in a
direction open to it and produces a three-dimerisiona]

150 THE FOURTTT DIMENSION

solid. Using the addition of* white to symbolise the
products of this motion this new kind of solid will be light
orange or ochre, and it will be bounded on the far side by
the final position of the orange square which traced it
out, and this final position we suppose to be coloured like
the square in its first position, orange with yellow and
red boundaries and null corners."

This product of movement, which it is so easy for us to
describe, would be difficult for him to conceive. But this
difficulty is connected rather with its totality than with
any particular part of it.

Any line, or plane of this, to him higher, solid we could
show to him, and put in his sensible world.

We have already seen how the pink square could be put
in his world by a turning of the cube about the red line.
And any section which we can conceive made of the cube
could be exhibited to him. You have simply to turn the
cube and push it through, so that the plane of his existence
is the j)lane which cuts out the given section of the cube,
then the section would appear to him as a solid. In his
world he would see the contour, get to any part of it by
digging down into it.

The Process by which a Plane Being would gain
a Notion of a Solid.

If we suppose the plane being to have a general idea of
the existence of a higher solid — our solid — we must next
trace out in detail the method, the discipline, by which
he would acquire a working familiarity with our space
existence. The process begins with an adequate realisa-
tion of a simple solid figure. For this purpose we will
suppose eight cubes forming a larger cube, and first we
will suppose each cube to be coloured throughout uniformly.

NOMENCLATURE AND ANALOGIES

151

Let the cubes in fig. 91 be the eight making a larger
cube.

Now, although each cube is supposed to be coloured
entirely through with the colour, the name of which is
written on it, still we can speak of the faces, edges, and
corners of each cube as if the colour scheme we have
investigated held for it. Thus, on the null cube we can
speak of a null point, a red line, a white line, a pink face, and
so on. These colour designations are shown on No. 1 of
the views of the tesseract in the plate. Here these colour

Orange-
Vellow

<fc

"Q>

Fig. 91.

names are used simply in their geometrical significance.
They denote what the particular line, etc., referred to would
have as its colour, if in reference to the particular cube
the colour scheme described previously were carried out.

If such a block of cubes were put against the plane and
then passed through it from right to left, at the rate of an
inch a minute, each cube being an inch each way, the
plane being would have the following appearances: —

First of all, four squares null, yellow, red, orange, Last ing
each a minute; and secondly, taking the exact places
of these four squares, four others, coloured white, lighl
yellow, pink, ochre. Tims, to make a catalogue of the
solid body, be would have 1<> put Bide by side in his world
two Bets of four square- each, a- iii lig. 92, The first

152

THE FOURTH DIMENSION

«r

Fig. 92.

are supposed to las! a minute, and then the others to

come in in place of them,

and also last a minute.
In speaking of them

he would have to denote
what part of the respective
cube each square repre-
sents. Thus, at the begin-
ning he would have null
cube orange face, and after
the motion had begun he
would have null cube ochre
section. As he could get
the same coloured section whichever way the cube passed
through, it would be best for him to call this section white
section, meaning that it is transverse to the white axis.
These colour-names, of course, are merely used as names,
and do not imply in this case that the object is really
coloured. Finally, after a minute, as the first cube was
passing beyond his plane he would have null cube orange
face again.

The same names will hold for each of the other cubes,
describing what face or section of them the plane being
has before him ; and the second wall of cubes will come
on, continue, and go out in the same manner. In the
area he thus has he can represent any movement which
we carry out in the cubes, as long as it does not involve
a motion in the direction of the white axis. The relation
of parts that succeed one another in the direction of the
white axis is realised by him as a consecution of states.

Now, his means of developing his space apprehension
lies in this, that that which is represented as a time
sequence in one position of the cubes, can become a real
co-existence, if something that has a real co-existence,
becomes a time sequence,

NOMENCLATURE AND ANALOGIES

153

"We must suppose the cubes turned round each of the
axes, the red line, and the yellow line, then something,
which was given as time before, will now be given as the
plane creature's space ; something, which was given as space
before, will now be given as a time series as the cube is
passed through the plane.

The three positions in which the cubes must be studied
are the one given above and the two following ones. In
each case the original null point which was nearest to us
at first is marked by an asterisk. In figs. 93 and 94 the

n
\

P-

och.

1. v.

Fig. 93.

The cube swung round the rod lino, so that the white line points

towards us.

point marked with a star is the same in the cubes and in
t he plane view.

In fig. 93 the cube is swung round the red line so as io
point towards as, and consequently the pink face conies
next to the plane. As it passes through there are hvo
varieties of appearance designated by the figures 1 and 2
in the plane. These appearances are named in the figure,
and are determined by the order in which the cubes

154

THE FOURTH MMKXBION

come in the motion of the whole block through the
plane.

With regard to these square- severally, however,
different names must be used, determined by their
relations in the block.

Thus, in fig. 93, when the cube first rests against the
plane the null cube is in contact by its pink face ; as the
block passes through we get an ochre section of the null
cube, but this is better called a yellow section, as it is
made by a plane perpendicular to the yellow line. When

*»

Fig. 94.

The cube swung round yellow line, with red line running from left
to right, and white line running down.

the null cube has passed through the plane, as it is
leaving it, we get again a pink face.

The same series of changes take place with the cube
appearances which follow on those of the null cube. In
this motion the yellow cube follows on the null cube, and
the square marked yellow in 2 in the plane will be first
" yellow pink face," then " yellow yellow section," then
" yellow pink face."

In fig. 94, in which the cube is turned about the yellow
line, we have a certain difficulty, for the plane being will

NOMENCLATURE AND ANALOGIES

155

find that the position his squares are to be placed in will
lie below that which they first occupied. They will come
where the support was on which he stood his first set of
squares. He will get over this difficulty by moving his
support.

Then, since the cubes come upon his plane by the light
yellow face, he will have, taking the null cube as before for
an example, null, light yellow face ; null, red section,
because the section is perpendicular to the red line ; and
finally, as the null cube leaves the plane, null, light yellow
face. Then, in this case red following on null, he will

Null
r. y. wh,

Null

3 4

Null
r. y. wh.

Fig. 95.

have the same series of views of the red as he had of the
null cube.

There is another set of considerations which we will
briefly allude to.

Suppose there is a hollow cube, and a string is stretched
across it from null to null, r, y, ich, as we may call the
far diagonal point, how will this string appear to the
plane being a- the cube moves transverse to his plane?

Lei u- represent the cube as a number of section-. Bay

5, corresponding to 1 equal divisions made along 1 lie while

line perpendicular <<> it .

We number these sections 0, 1, 2, ?>, 4, corresponding

1<> the distances along the white lino at which they are

1 56

TIIK ForilTII DIMENSION

taken, and imagine each section to come in successively,
taking the place of the preceding one.

These sections appear to the plane being, counting from

the first, to exactly coincide each with the preceding one.
But the section of the string occupies a different place in
each to that which it does in the preceding section. The
section of the string appears in the position marked by
the dots. Hence the slant of the string appears as a
motion in the frame work marked out by the cube sides.
If we suppose the motion of the cube not to be recognised,
then the string appears to the plane being as a moving
point. Hence extension on the unknown dimension
appears as duration. Extension sloping in the unknown
direction appears as continuous movement.

CHAPTER XII

THE SIMPLEST FOUR-DIMENSIONAL SOLID

A PLANE being, in learning to apprehend solid existence,
must first of all realise that there is a sense of direction
altogether wanting to him. That which we call right
and left does not exist in his perception. He must
assume a movement in a direction, and a distinction of
positive and negative in that direction, which has no
reality corresponding to it in the movements he can
make. This direction, this new dimension, he can only
make sensible to himself by bringing in time, and sup-
posing that changes, which take place in time, are due to
objects of a definite configuration in three dimensions
passing transverse to his plane, and the different sections
of it being apprehended as changes of one and the same
plane figure.

He must also acquire a distinct notion about his plane
world, he must no longer believe that it is the all of
space, but that space extends on both sides of it. In
order, then, to prevent his moving off in this unknown
direction, lie must assume a sheet, an extended solid sheet,
in two dimensions, against which, in contact with which,
all bis movement a take place.

When we come to think of a four-dimensional solid,
what are the corresponding assumptions which we most
make ?

We most suppose a sense which we have Dot, a sense

K>s THE F0URTI1 DIMENSION

of direction wanting in us, something which a being in
a four-dimensional world has, and which we have Dot. Jt

is a sense corresponding to a new space direction, a
direction which extends positively and negatively from
every point of our space, and which goes right away from
any space direction we know of. The perpendicular to a
plane is perpendicular, not only to two lines in it, but to
every line, and so we must conceive this fourth dimension
as running perpendicularly to each and every line we can
draw in our space.

And as the plane being had to suppose something
which prevented his moving off in the third, the
unknown dimension to him, so we have to suppose
something which prevents us moving off in the direction
unknown to us. This something, since we must be in
contact with it in every one of our movements, must not
be a plane surface, but a solid ; it must be a solid, which
in every one of our movements we are against, not in. It
must be supposed as stretching out in every space dimension
that we know ; but we are not in it, we are against it, we
are next to it, in the fourth dimension.

That is, as the plane being conceives himself as having
a very small thickness in the third dimension, of which
he is not awTare in his sense experience, so we must
suppose ourselves as having a very small thickness in
the fourth dimension, and, being thus four-dimensional
beings, to be prevented from realising that we are
such beings by a constraint which keeps us always in
contact with a vast solid sheet, which stretches on in
every direction. We are against that sheet, so that, if we
had the power of four-dimensional movement, we should
either go away from it or through it ; all our space
movements as we know them being such that, performing
them, we keep in contact with this solid sheet.

Now consider the exposition a plane being would make

THE SIMPLEST FOUR-DIMENSIONAL SOLID

159

Fig. 9G.

for himself as to the. question of the enclosure of a square,
and of a cube.

He would say the square A, in Fig. 96, is completely
enclosed by the four squares, A far,
A near, A above, A below, or as they
are written kn, a/, Act, kb.

If now he conceives the square A
to move in the, to him, unknown
dimension it will trace out a cube,
and the bounding squares will form
cubes. Will these completely sur-
round the cube generated by A ? Jso ;
there will be two faces of the cube
made by A left uncovered; the first,
that face which coincides with the
square A in its first position ; the next, that which coincides
with the square A in its final position. Against these
two faces cubes must be placed in order to completely
enclose the cube A. These may be called the cubes left
and right or kl and a?\ Thus each of the enclosing
squares of the square A becomes a cube and two more
cubes are wanted to enclose the cube formed by the
movement of A in the third dimension.

The plane being could not see the square A with the
squares An, a/, etc., placed about it,
because they completely hide it from
view ; and so we, in the analogous
case in our three-dimensional world,
cannot see a cube A surrounded by
six other cubes. These cubes we
will call a near \n, a far a/, a above
Aa, a below a//, a left a/, a right .\/\
shown in fig. 97. If now the cube A
moves in tin' fourth dimension righl oul of space, it traces
out a higher cube — a tesseract, ;i- it mav be called.

Fit:. :»:.

100 THE FOURTH DIMENSION

Each of the six surrounding cubes carried on in the same
motion will make a tesserae! also, and these will be
grouped around the tesseract formed by A. But will they
enclose it completely ?

All the cubes An, a/, etc., lie in our space. But there is
nothing between the cube A and that solid sheet in contact
with which every particle of matter is. When the cube A
moves in the fourth direction it starts from its position,
say Ah, and ends in a final position An (using the words
" ana " and " kata " for up and down in the fourth dimen-
sion). Now the movement in this fourth dimension is
not bounded by any of the cubes An, a/, nor by what
they form when thus moved. The tesseract which a
becomes is bounded in the positive and negative ways in
this new direction by the first position of A and the last
position of A. Or, if we ask how many tesseract s lie
around the tesseract which A forms, there are eight, of
which one meets it by the cube a, and another meets it
by a cube like A at the end of its motion.

We come here to a very curious thing. The whole
solid cube A is to be looked on merely as a boundary of
the tesseract.

Yet this is exactly analogous to what the plane being
would come to in his study of the solid world. The
square A (fig. 96), which the plane being looks on as a
solid existence in his plane world, is merely the boundary
of the cube which he supposes generated by its motion.

The fact is that we have to recognise that, if there is
another dimension of space, our present idea of a solid
body, as one which has three dimensions only, does not
correspond to anything real, but is the abstract idea of a
three-dimensional boundary limiting a four-dimensional
solid, which a four-dimensional being would form. The
plane being's thought of a square is not the thought
of what we should call a possibly existing real square,

THE SIMPLEST FOUR-DlMENSIONAL SOLTD

161

but the thought of an abstract boundary, the face of
a cube.

Let us now take our eight coloured cubes, which form
a cube in space, and ask what additions we must make
to them to represent the simplest-collection of four-dimen-
sional bodies — namely, a group of them of the same extent
in every direction. In plane space we have four squares.
In solid space we have eight cubes. So we should expect
in four-dimensional space to have sixteen four-dimen-
sional bodies — bodies which in four-dimensional space
correspond to cubes in three-dimensional space, and these
bodies we call tesseracts.

Given then the null, white, red, yellow cubes, and
those which make up the block, we
notice that we represent perfectly
well the extension in three directions
(fig. 98). From the null point of
the null cube, travelling one inch, we
come to the white cube ; travelling
one inch away we come to the yellow
cube ; travelling one inch up we come
to the red cube. Now, if there is
a fourth dimension, then travelling
from the same null point for one
inch in that direction, we must come to the body lying
beyond the null region.

I say null region, not cube ; for with the introduction
of the fourth dimension each of our cubes must become
something different from cubes. If they are to have
existence in the fourth dimension, they must be " filled
up from" in 1 liis fourth dimension.

Now we will assume that as we get a transference from
null to while going in one way, from null to yellow going
in another, so going from null in the fourth direction we
have a transference from null to blue, using thus the

Red

Pink

Null

White

, yellow

(Orange hidden)
Fin-. 98.

102

T1IK I'oriM'H DIMENSION

colours while, yellow, red, blue, to denote transferences in
each of the four directions — right, away, up, unknown or
fourth dimension.

Hence, as the plane being must represent the solid re-
gions, he would come to by going right, as four squares Lying

in some position in
his plane, arbitrarily
chosen, side by side
with his original four
squares, so we must
represent those eight
four-dimensional re-
gions, which we
Fig. 99. should come to by

A plane being's representation of a block croino" in the fourth

of eight cubes by two sets of four squares. j:_ • r _ i

b J 1 dimension from each

of our eight cubes, by eight cubes placed in some arbitrary

position relative to our first eight cubes.

Red

Pink

Null

White

low\ Light
vYellow

Purple

Light
purple

Blue

Light
blue

-4 Light
brown

Green

TlihT

green

(1)
Orange hidden

(2)
Brown hidden

Fig. 100.

Our representation of a block of sixteen tesseracts by
two blocks of eight cubes.*

Hence, of the two sets of eight cubes, each one will serve

* The eight cubes used here in 2 can be found in the second of the
tnodcl blocks. They can be taken out and used.

THE SIMPLEST FOUK-DIMENSIONAL SOLID 163

us as a representation of one of the sixteen tesseracts
which form one single block in four-dimensional space.
Each cube, as we have it, is a tray, as it were, against
which the real four-dimensional figure rests — just as each
of the squares which the plane being has is a tray, so to
speak, against which the cube it represents could rest.

If we suppose the cubes to be one inch each way, then
the original eight cubes will give eight tesseracts of the
same colours, or the cubes, extending each one inch in the
fourth dimension.

But after these there come, going on in the fourth di-
mension, eight other bodies, eight other tesseracts. These
must be there, if we suppose the four-dimensional body
we make up to have two divisions, one inch each in each
of four directions.

The colour we choose to designate the transference to
this second region in the fourth dimension is blue. Thus,
starting from the null cube and going in the fourth
dimension, we first go through one inch of the null
tesseract, then we come to a blue cube, which is the
beginning of a blue tesseract. This blue tesseract stretches
one inch farther on in the fourth dimension.

Thus, beyond each of the eight tesseracts, which are of
the same colour as the cubes which are their bases, lie
eight tesseracts whose colours are derived from the colours
of the first eight by adding blue. Thus —
Null gives blue

Yellow „ green

Red „ purple

Orange „ brown

White „ light blue

Pink „ light purple

Lighl yellow ,, light green
( >chre „ lighl brown

The addition of blue bo yellow gives green — this \a a

104 THE FOURTH DIMENSION

natural supposition to make. It is also natural to suppose
that blue added to red makes purple. Orange and blue
can be made to give a brown, by using certain shades and

proportions. And ochre and blue can be made to give a
light brown.

But the scheme of colours is merely used for getting
a definite and realisable set of names and distinctions
visible to the eye. Their naturalness is apparent to any
one in the habit of using colours, and may be assumed to
be justifiable, as the sole purpose is to devise a set of
names which are easy to remember, and which will give
us a set of colours by which diagrams may be made easy
of comprehension. No scientific classification of colours
has been attempted.

Starting, then, with these sixteen colour names, we have
a catalogue of the sixteen tesseracts, which form a four-
dimensional block analogous to the cubic block. But
the cube wThich we can put in space and look at is not one
of the constituent tesseracts ; it is merely the beginning,
the solid face, the side, the aspect, of a tesseract.

We will now proceed to derive a name for each region,
point, edge, plane face, solid and a face of the tesseract.

The system will be clear, if we look at a representation
in the plane of a tesseract with three, and one with four
divisions in its side.

The tesseract made up of three tesseracts each way
corresponds to the cube made up of three cubes each way,
and will give us a complete nomenclature.

In this diagram, fig. 101, 1 represents a cube of 27
cubes, each of which is the beginning of a tesseract.
These cubes are represented simply by their lowest squares,
the solid content must be understood. 2 represents the
27 cubes which are the beginnings of the 27 tesseracts
one inch on in the fourth dimension. These tesseracts
are represented as a block of cubes put side by side with

THE SIMPLEST FOUR-DIMENSIONAL SOLID

165

the first block, but in their proper positions they could
not be in space with the first set. 3 represents 27 cubes

Fie:. 101.

Null

White

Null

bellow

Light
yellow

Yellow

Null

White

Null

Blue

Light
blue

Blue

Green

Light
green

Green

Blue

Light
blue

Blue

Null

Yellow

Null

White

Light
yellow

White

Null

Yellow

NuR

Red

Pink

Red

Orange

Ochre

)range

Red

Pink

Red

Purple

Light
purple

Purple

Brown

Light
brown

Brown

Purple

Light
purple

Purple

Red

Pink

Red

)range

Ochre

) range

Red

Pink

Red

Null

White

Null

Blue

Light
blue

Blue

Null

White

Null

Yellow

Light
yellow

Yellow

Green

Light
green

Green

Yellow

Light
yellow

Yellow

Null

White

Null

Blue

Light
blue

Blue

Null

White

Null

Bach cube is the begin-
niog of tb • i ici

going in the fourth di-
men&ion.

Each cube fa the begin-
ning of the leoond
tesseract.

Each cube is tho begin-
ning of the third
teaseract.

(forming a larger cube) which are the beginnings of the
tesseracts, which begin two Inches in the fourth direction
from our -pace and continue another inch,

166

THE FOURTTT DIMENSION

In fig. 102, we have the representation of a block of
4 x 4 x 4 x 4 or 2.~>(> tesseracts. They are given in

Fig. 102.*

i*"

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X)'

X>'

x>
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A cube of 64 cubes,
each 1 in. x 1 in.
x 1 in., the begin-
u ins of a tesseract.

j*-

<••

x>-

x>
V

x>'

x>-

x>J

•v

<^-

N>-

<^-

v^1

•v

s-"*

N>"

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<>•

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v^-

A cubo of 04 cubes,
each 1 in. x 1 in.
x 1 iu. the begin-
ning of tesseracts
1 in. from our space
on the 4th dimen-
sion.

A cubo of C<4 cubes,

each 1 in. x 1 in.
x 1 in., the begin-
ning of tesseracts
2 in. from our space
in the 4th dimen-
sion.

A cube of G4 cubes,
each 1 in. x 1 in.
x 1 in., the begin-
ning of tesseracts
Btaxting 3 in. from
our space in the 4th
dimension.

four consecutive sections, each supposed to be taken one
inch apart in the fourth dimension, and so giving four

* The coloured plate, figs, 1, 2, 3, shows these relations more
conspicuously.

THE SIMPLEST FOUR-DIMENSIONAL SOLID 107

blocks of cubes, 64 in each block. Here we see, com-
paring it with the figure of 81 tesseracts, that the number
of the different regions show a different tendency of
increase. By taking five blocks of five divisions each way
this would become even more clear.

We see, fig. 102, that starting from the point at any
corner, the white coloured regions only extend out in
a line. The same is true for the yellow, red, and blue.
With regard to the latter it should be noticed that the
line of blues does not consist in regions next to each
other in the drawing, but in portions which come in in
different cubes. The portions which lie next to one
another in the fourth dimension must always be repre-
sented so, when we have a three-dimensional representation.
Again, those regions such as the pink one, go on increasing
in two dimensions. About the pink region this is seen
without going out of the cube itself, the pink regions
increase in length and height, but in no other dimension.
In examining these regions it is sufficient to take one as
a sample.

The purple increases in the same manner, for it comes
in in a succession from below to above in block 2, and in
a succession from block to block in 2 and 3. Now, a
succession from below to above represents a continuous
extension upwards, and a succession from block to block
represents a continuous extension in the fourth dimension.
Thus the purple regions increase in two dimensions, the
upward and the fourth, so when we take a very great
many divisions, and let each become very small, the
purple region forms a I wo-dimensional extension.

In the same way, looking at the regions marked 1. b. or

lighi blue, which starts nearest a corner, we see that the

eracts occupying it increase in length from left to

right, forming a line and that there are as many lines of

light blii<' tesseracts as there are sections between the

108 THE FOURTH DIMENSION

first and last section. Hence the light blue tesseractfl
increase in number in two ways — in the right and Left,
and in the fourth dimension. They ultimately form
what we may call a plane surface.

Now all those regions which contain a mixture of two
simple colours, white, yellow, red, blue, increase in two
ways. On the other hand, those which contain a mixture
of three colours increase in three ways. Take, for instance,
the ochre region ; this has three colours, white, yellow,
red ; and in the cube itself it increases in three ways.

Now regard the orange region ; if we add blue to this
we get a brown. The region of the brown tesseracts
extends in two ways on the left of the second block,
No. 2 in the figure. It extends also from left to right in
succession from one section to another, from section 2
to section 3 in our figure.

Hence the brown tesseracts increase in number in three
dimensions upwards, to and fro, fourth dimension. Hence
they form a cubic, a three-dimensional region; this region
extends up and down, near and far, and in the fourth
direction, but is thin in the direction from left to right.
It is a cube which, when the complete tesseract is repre-
sented in our space, appears as a series of faces on the
successive cubic sections of the tesseract. Compare fig.
103 in which the middle block, 2, stands as representing a
great number of sections intermediate between 1 and 3.

In a similar way from the pink region by addition of
blue we have the light purple region, which can be seen
to increase in three ways as the number of divisions
becomes greater. The three ways in which this region of
tesseracts extends is up and down, right and left, fourth
dimension. Finally, therefore, it forms a cubic mass of
very small tesseracts, and when the tesseract is given in
space sections it appears on the feces containing the
upward and the right and left dimensions,

THE SIMPLEST FOUR-DIMENSIONAL SOLID 109

We get then altogether, as three-dimensional regions,
ochre, brown, light purple, light green.

Finally, there is the region which corresponds to a
mixture of all the colours ; there is only one region such
as this. It is the one that springs from ochre by the
addition of blue — this colour we call light brown.

Looking at the light brown region we see that it
increases in four ways. Hence, the tesseracts of which it
is composed increase in number in each of four dimen-
sions, and the shape they form does not remain thin in
any of the four dimensions. Consequently this region
becomes the solid content of the block of tesseracts, itself;
it is the real four-dimensional solid. All the other regions
are then boundaries of this light brown region. If we
suppose the process of increasing the number of tesseracts
and diminishing their size carried on indefinitely, then
the light brown coloured tesseracts become the whole
interior mass, the three-coloured tesseracts become three-
dimensional boundaries, thin in one dimension, and form
the ochre, the brown, the light blue, the light green.
The two-coloured tesseracts become two-dimensional
boundaries, thin in two dimensions, e.g.f the pink, the
green, the purple, the orange, the light blue, the light
yellow. The one-coloured tesseracts become bounding
lines, thin in three dimensions, and the null points become
bounding corners, thin in four dimensions. From these
thin real boundaries we can pass in thought to the
abstractions — points, lines, faces, solids — bounding the
four-dimensional solid, which is this case is light brown
coloured, and under this supposition the light brown
coloured region is the only real one, is the only one which
is not ;m abstraction.

It Bhould be observed that, in taking a Bquare as the
representation of a cube on a plane, we only represent
one face, or the -action between two faces. The squares,

170 TITE FOURTTT DIMENSION

as drawn by a plane being, are not the cubes themsel
but represent the laces or the sections of a cube. Thus
in the plane being's diagram a cube of twenty-seven cubes

"null" represents a cube, but is really, in the normal
position, the orange square of a null cube, and may be
called null, orange square.

A plane being would save himself confusion if he named
his representative squares, not by using the names of the
cubes simply, but by adding to the names of the cubes a
word to show what part of a cube his representative square
was.

Thus a cube null standing against his plane touches it
by null orange face, passing through his plane it has in
the plane a square as trace, which is null white section, if
we use the phrase white section to mean a section drawn
perpendicular to the white line. In the same way the
cubes which we take as representative of the tesseract are
not the tesseract itself, but definite faces or sections of it.
In the preceding figures we should say then, not null, but
" null tesseract ochre cube," because the cube we actually
have is the one determined by the three axes, white, red,
yellow.

There is another way in which we can regard the colour
nomenclature of the boundaries of a tesseract.

Consider a null point to move tracing out a white line
one inch in length, and terminating in a null point,
see fig. 103 or in the coloured plate.

Then consider this white line with its terminal points
itself to move in a second dimension, each of the points
traces out a line, the line itself traces out an area, and
gives two lines as well, its initial and its final position.

Thus, if we call "a region" any element of the figure,
such as a point, or a line, etc., every "region" in moving
traces out a new kind of region, " a higher region," and
gives two regions of its own kind, an initial and a final

THE SIMPLEST FOUR-DIMENSIONAL SOLID 171

position. The " higher region " means a region with
another dimension in it.

Now the square can move and generate a cube. The
square light yellow moves and traces out the mass of the
cube. Letting the addition of red denote the region
made by the motion in the upward direction we get an
ochre solid. The light yellow face in its initial and
terminal positions give the two square boundaries of the
cube above and below. Then each of the four lines of the
light yellow square — white, yellow, and the white, yellow
opposite them — trace out a bounding square. So there
are in all six bounding squares, four of these squares being
designated in colour by adding red to the colour of the
generating lines. Finally, each point moving in the up
direction gives rise to a line coloured null + red, or red,
and then there are the initial and terminal positions of the
points giving eight points. The number of the lines is
evidently twelve, for the four lines of this light yellow
square give four lines in their initial, four lines in their
final position, while the four points trace out four lines,
that is altogether twelve lines.

Now the squares are each of them separate boundaries
of the cube, while the lines belong, each of them, to two
Bquares, thus the red line is that which is common to the
orange and pink squares.

Now suppose that there is a direction, the fourth
dimension, which is perpendicular alike to every one
of the space dimensions already used — a dimension
perpendicular, for instance, to up and to right hand,
so that the pink square moving in this direction traces
out a cube.

A dimension, moreover, perpendicular to the up and
away directions, bo thai the orange square moving in tin's
direction also traces ou1 a cube, and the lighl yellow
square, too, moving in this direct inn traces out a cube,

172 THE FOURTH DIMENSION

Under this supposition, the whole cube moving in the
unknown dimension, traces out something new — a new-
kind of volume, a higher volume. This higher volume
is a four-dimensional volume, and we designate it in colour
by adding blue to the colour of that which by moving
generates it.

It is generated by the motion of the ochre solid, and
hence it is of the colour we call light brown (white, yellow,
red, blue, mixed together). It is represented by a number
of sections like 2 in fig. 103.

Now this light browrn higher solid has for boundaries :
first, the ochre cube in its initial position, second, the
same cube in its final position, 1 and 3, fig. 103. Each
of the squares which bound the cube, moreover, by move-
ment in this new direction traces out a cube, so we have
from the front pink faces of the cube, third, a pink blue or
light purple cube, shown as a light purple face on cube 2
in fig. 103, this cube standing for any number of inter-
mediate sections ; fourth, a similar cube from the opposite
pink face ; fifth, a cube traced out by the orange face —
this is coloured brown and is represented by the brown
face of the section cube in fig. 103 ; sixth, a correspond-
ing brown cube on the right hand ; seventh, a cube
starting from the light yellow square belowT ; the unknown
dimension is at right angles to this also. This cube is
coloured light yellow^ and blue or light green ; and,
finally, eighth, a corresponding cube from the upper
light yellow face, shown as the light green square at the
top of the section cube.

The tesseract has thus eight cubic boundaries. These
completely enclose it, so that it would be invisible to a
four-dimensional being. Now, as to the other boundaries,
just as the cube has squares, lines, points, as boundaries,
so the tesseract has cubes, squares, lines, points, as
boundaries,

THE SIMPLEST FOUR-DIMENSIONAL SOLID 173

The number of squares is found thus — round the cube
are six squares, these will give six squares in their initial
and six in their final positions. Then each of the eight
lines of the cube trace out a square in the motion in the
fourth dimension. Hence there will be altogether
12 + 8 = 20 squares.

If we look at any one of these squares we see that it
is the meeting surface of two of the cubic sides. Thus,
the red line by its movement in the fourth dimension,
traces out a purple square — this is common to two
cubes, one of which is traced out by the pink square
moving in the fourth dimension, and the other is
traced out by the orange square moving in the same
way. To take another square, the light yellow one, this
is common to the ochre cube and the light green cube.
The ochre cube comes from the light yellow square
by moving it in the up direction, the light green cube
is made from the light yellow square by moving it in
the fourth dimension. The number of lines is thirty-
two, for the twelve lines of the cube give twelve lines
of the tesseract in their initial position, and twelve in
their final position, making twenty-four, while each of
the eight points traces out a line, thus forming thirty -
two lines altogether.

The lines are each of them common to three cubes, or
to three square faces ; take, for instance, the red line.
This is common to the orange face, the pink face, and
that face which is formed by moving the red line in the
sixth dimension, namely, the purple face. It is also
common to the ochre cube, the pale purple cube, and the
brown cube.

The points are common to six square faces and to four
cubes ; thus, the null point from which we start is common
to the three -quare faces — pink, light yellow, orange, and
to the three square feces made by moving the three lines

174

THE l<>r kill DIMENSION

white, yellow, red, in the fourth dimension, namely, the
Light blue, the light green, the purple faces — that is, to
six faces in all. The four cubes which meet in it are the

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A complete view of the tesseract in its various space

THE SIMPLEST FOUR-DIMENSIONAL SOLID

175

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17G

T1IK Fnl'KTll DIM KNSION

represents the view of a tesseract coloured as described as
it begins to pass transverse to our space. The intermedial e
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passing out. These figures will be explained in detail in
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THE SIMPLEST FOUR-DIMENSIONAL SOLID

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We have thus obtained a nomenclature for each of the
regions of a tesseract ; we can speak of any one of the
eight bounding cubes, the twenty square faces, the thirty-
two lines, the sixteen points.

CHAPTER XIII

REMARKS ON THE FIGURES

An inspection of above figures will give an answer to
many questions about the tesseract. If we have a
tesseract one inch each way, then it can be represented
as a cube — a cube having white, yellow, red axes, and
from this cube as a beginning, a volume extending into
the fourth dimension. Now suppose the tesseract to pass
transverse to our space, the cube of the red, yellow, white
axes disappears at once, it is indefinitely thin in the
fourth dimension. Its place is occupied by those parts
of the tesseract which lie further away from our space
in the fourth dimension. Each one of these sections
will last only for one moment, but the whole of them
will take up some appreciable time in passing. If we
take the rate of one inch a minute the sections will take
the whole of the minute in their passage across our
space, they will take the whole of the minute except the
moment which the beginning cube and the end cube
occupy in their crossing our space. In each one of the
cubes, the section cubes, we can draw lines in all directions
except in the direction occupied by the blue line, the
fourth dimension ; lines in that direction are represented
by the transition from one section cube to another. Thus
to give ourselves an adequate representation of the
tesseract we ought to have a limitless number of section
cubes intermediate between the first bounding cube, the

17S

REMARKS ON THE FIGURES

179

ochre cube, and the last bounding cube, the other ochre
cube. Practically three intermediate sectional cubes will
be found sufficient for most purposes. We will take then
a series of five figures — two terminal cubes, and three
intermediate sections — and show how the different regions
appear in our space when we take each set of three out
of the four axes of the tesseract as lying in our space.

In fig. 107 initial letters are used for the colours.
A reference to fig. 103 will show the complete nomen-
clature, which is merely indicated here.

l.,\

n- wh. n- bl. 1. bl. bl. *bl. 1. bl. bl. 'bl. 1. bl. bl. " n. wh n.
interior interior interior interior interior

Ochre L.Brown L.Brown L.Brown

Fig. 107.

Ochre

In this figure the tesseract is shown in five stages
distant from our space: first, zero ; second, J in.; third,
| in. ; fourth, f in. ; fifth, 1 in.; which are called 60, 61,
62, 63, 64, because they are sections taken at distances
0, 1, 2, 3, 4 quarter inches along the blue line. All the
regions can be named from the first cube, the 60 cube,
as before, simply by remembering that transference along
the 6 axis gives the addition of blue to the colour of
the region in the ochre, the 60 cube. In the final cube
64, the colouring of the original 60 cube is repeated.
Thua the red line moved along the blue axis gives a red
and blue or purple square. This purple square appears
as the three purple lines in the sections 61, 62, 63, taken
at J, ';, J of an inch in the fourth dimension. If the
tesseract moves transverse to our space we have then in
this particular region, first of all a red line which lasts
for a moment, secondly a purple line which takes its

180 THE FOURTH DIMENSION

place. This purple line lasts for a minute — that is, all
of a minute, except the moment taken by the crossing
our space of the initial and final red line. The purple
line having lasted for this period is succeeded by a red
line, which lasts for a moment; then this goes and the
tesseract has passed across our space. The final red line
we call red bl., because it is separated from the initial
red line by a distance along the axis for which we use
the colour blue. Thus a line that lasts represents an
area duration ; is in this mode of presentation equivalent
to a dimension of space. In the same way the white
line, during the crossing our space by the tesseract, is
succeeded by a light blue line which lasts for the inside
of a minute, and as the tesseract leaves our space, having
crossed it, the white bl. line appears as the final
termination.

Take now the pink face. Moved in the blue direction
it traces out a light purple cube. This light purple
cube is shown in sections in bu 62, b3, and the farther
face of this cube in the blue direction is shown in 64 —
a pink face, called pink b because it is distant from the
pink face we began with in the blue direction. Thus
the cube which we colour light purple appears as a lasting
square. The square face itself, the pink face, vanishes
instantly the tesseract begins to move, but the light
purple cube appears as a lasting square. Here also
duration is the equivalent of a dimension of space — a
lasting square is a cube. It is useful to connect these
diagrams with the views given in the coloured plate.

Take again the orange face, that determined by the
red and yellow axes ; from it goes a brown cube in the
blue direction, for red and yellow and blue are supposed
to make brown. This brown cube is shown in three
sections in the faces 6l? o2> b3. In 64 is the opposite
orange face of the brown cube, the face called orange b,

REMARKS ON THE FIGURES

181

for it is distant in the blue direction from the orange
face. As the tesseract passes transverse to our space,
we have then in this region an instantly vanishing orange
square, followed by a lasting brown square, and finally
an orange face which vanishes instantly.

-Now, as any three axes will be in our space, let us send
the white axis out into the unknown, the fourth dimen-
sion, and take the blue axis into our known space
dimension. Since the white and blue axes are perpen-
dicular to each other, if the white axis goes out into
the fourth dimension in the positive sense, the blue axis
will come into the direction the white axis occupied,
in the negative sense.

wh.

wh,

•

1- pur. ?

n. bl. n

1. bl.wl

j^X^*

1. pur. >

ur.

I. bl.wh. 1. bl. wh. n. bl. n.
Fig. 108.

Hence, not to complicate matters by having to think
of two senses in the unknown direction, let us send the
white line into the positive sense of the fourth dimen-
sion, and take the blue one as running in the negative
sense of that direction which the white line has left ;
let the blue line, that is, run to the left. We have
now the row of figures in fig. 108. The dotted cube
si lows where we had a cube when the white line ran
in our space — nowT it has turned out of our space, and
another solid boundary, another cubic face of the tesseract
comes into our space This cube has red and yellow
axes as before; but now, instead of a white axis running
to the right, there is a blue axis running to the left.
Here we can distinguish the regions by colours in a per-
fectly systematic way. The red line traces out a purple

JN'J THE FOUBTH DIMENSION

square in the transference along the blue axis by which

this cube is generated from the orange face. This
purple square made by the motion of the red line is
the same purple face that we saw before as a series of
lines in the sections blt b2, b:). Here, since both red and
blue axes are in our space, we have no need of duration
to represent the area they determine. In the motion
of the tesseract across space this purple face would
instantly disappear.

From the orange face, which is common to the initial
cubes in fig. 107 and fig. 108, there goes in the blue
direction a cube coloured brown. This brown cube is
now all in our space, because each of its three axes run
in space directions, up, away, to the left. It is the same
brown cube which appeared as the successive faces on the
sections bu b2} b:i. Having all its three axes in our
space, it is given in extension ; no part of it needs to
be represented as a succession. The tesseract is now
in a new position with regard to our space, and when
it moves across our space the brown cube instantly
disappears.

In order to exhibit the other regions of the tesseract
we must remember that now the white line runs in the
unknown dimension. Where shall we put the sections
at distances along the line ? Any arbitrary position in
our space will do : there is no way by which we can
represent their real- position.

However, as the brown cube comes off from the orange
face to the left, let us put these successive sections to
the left. We can call them tvh0, whu wh2, ivh3, whA,
because they are sections along the white axis, which
now runs in the unknown dimension.

Running from the purple square in the white direction
we find the light purple cube. This is represented in the
sections whu tuh2, ivh3, ivhA, fig. 108. It is the same cube

REMARKS ON THE FIGURES

183

that is represented in the sections bu b2, bz\ in fig. 107
the red and white axes are in our space, the blue out of
it ; in the other case, the red and blue are in our space,
the white out of it. It is evident that the face pink y,
opposite the pink face in fig. 107, makes a cube shown
in squares in b* 6* 63, K on the opposite side to the I
purple squares. Also the light yellow face at the base

of the cube 60, makes a liSU Sreen cube' 8ll0Wn as a serieS
of base squares.

The same light green cube can be found in fig. 107.
The base square in who is a green square, for it is enclosed
by blue and yellow axes. From it goes a cube in the
white direction, this is then a light green cube and the
same as the one just mentioned as existing in the sections

fy>, &i, b2, 63, bA. .

The case is, however, a little different with the brown
cube. This cube we : have altogether in space in the
section ivh01 fig. 108, while it exists as a series of squares,
the left-hand ones, in the sections b0, bv b,, b3, 64. The
brown cube exists as a solid in our space, as shown in
fig. 108. In the mode of representation of the tesseract
exhibited in fig. 107, the same brown cube appears as a
succession of squares. That is, as the tesseract moves
across space, the brown cube would actually be to us a
square— it would be merely the lasting boundary of another
solid. It would have no thickness at all, only extension
in two dimensions, and its duration would show its solidity
in three dimensions.

It is obvious that, if there is a four-dimensional space,
matter in three dimensions only is a mere abstraction ; all
material objects must then have a slight four-dimensional
thickness. In this case the above statement will undergo
modification. The material cube which is used as the
model of the boundary of a tesseract will bave a slight
thickness in the fourth dimension, and when the cube is

I 84 THE FOURTH DIMENSION

presented fcb us in another aspect, it would not be a mere
surface. Bnt it is most convenient to regard the cubes
we use as having no extension at all in the fourth
dimension. This consideration serves to bring out a point
alluded to before, that, if there is a fourth dimension, our
conception of a solid is the conception of a mere abstraction,
and our talking about real three-dimensional objects would
seem to a four-dimensional being as incorrect as a two-
dimensional being's telling about real squares, real
triangles, etc., would seem to us.

The consideration of the two views of the brown cube
shows that any section of a cube can be looked at by a
presentation of the cube in a different position in four-
dimensional space. The brown faces in o1? b2, b3, are the
very same brown sections that would be obtained by
cutting the browm cube, ivh0, across at the right distances
along the blue line, as shown in fig. 108. But as these
sections are placed in the brown cube, who, they come
behind one another in the blue direction. Now, in the
sections ivhls ivh2, tvh3} we are looking at these sections
from the white direction — the blue direction does not
exist in these figures. So we see them in a direction at
right angles to that in which they occur behind one
another in tvh0. There are intermediate views, which
would come in the rotation of a tesseract. These brown
squares can be looked at from directions intermediate
between the white and blue axes. It must be remembered
that the fourth dimension is perpendicular equally to all
three space axes. Hence wTe must take the combinations
of the blue axis, with each two of our three axes, white,
red, yellow, in turn.

In fig. 109 wre take red, white, and blue axes in space,
sending yellow into the fourth dimension. If it goes into
the positive sense of the fourth dimension the blue line
will come in the opposite direction to that in which the

REMARKS ON THE FIGURES

185

yellow line ran before. Hence, the cube determined by
the white, red, blue axes, will start from the pink plane
and run towards us. The dotted cube shows where the
ochre cube was. When it is turned out of space, the cube
coming towards from its front face is the one which comes
into our space in this turning. Since the yellow line now
runs in the unknown dimension we call the sections
7/0, yly y2, 2/3. 2/4, as they are made at distances 0, 1, 2, 3, 4,
quarter inches along the yellow line. We suppose these
cubes arranged in a line coming towards us — not that
that is any more natural than any other arbitrary series
of positions, but it agrees with the plan previously adopted.

y0 y* y2 y3 y4
%

och.

gr-

yv,

och.

l.bl?

«%

Fig. 109.

The interior of the first cube, y0, is that derived from
pink by adding blue, or, as we call it, light purple. The
faces of the cube are light blue, purple, pink. As drawn,
we can only see the face nearest to us, which is not the
one from which the cube starts — but the face on the
opposite side has the same colour name as the face
towards us.

The successive sections of the series, y0, 7/„ ?/,, etc., can
be considered as derived from seel ions of the b0 cube
made at distances along the yellow axis. What is distant
a quarter inch from the pink face in the yellow direct ion ?
This question is answered by taking a section from a point
a quart er inch along 1 1m1 yellow axis in the cube fy„ tig. 107.
It 1- ;m ochre section with lines orange and light yellow.
This r-ection will therefore take the place of the pink lace

180

THE FOURTH 1H.MKNS10N

in vy, when we go on in the yellow direction. Tims, the
first section, ?/,, will begin from an ochre face with light
yellow and orange lines. The colour of the axis which
lies in space towards us is blue, hence the regions of this
section-cube are determined in nomenclature, they will be
found in full in fig. 105.

There remains only one figure to be drawn, and that is
the one in which the red axis is replaced by the blue.
Here, as before, if the red axis goes out into the positive
sense of the fourth dimension, the blue line must come
into our space in the negative sense of the direction which
the red line has left. Accordingly, the first cube will

Fig. 110.

come in beneath the position of our ochre cube, the one
we have been in the habit of starting with.

To show these figures we must suppose the ochre cube
to be on a movable stand. When the red line swings out
into the unknown dimension, and the blue line comes in
downwards, a cube appears below the place occupied by
the ochre cube. The dotted cube shows where the ochre
cube was. That cube has gone and a different cube runs
downwards from its base. This cube has white, yellow,
and blue axes. Its top is a light yellow square, and hence
its interior is light yellow + blue or light green. Its front
face is formed by the white line moving along the blue
axis, and is therefore light blue, the left-hand side is
formed by the yellow line moving along the blue axis, and
therefore green.

REMAKES ON THE FIGURES 187

As the red line now runs in the fourth dimension, the
successive sections can be called ro, ri, 9*2, Tz, n, these
letters indicating that at distances 0, £, f, f, 1 inch along
the red axis we take all of the tesseract that can be fouud
in a three-dimensional space, this three-dimensional space
extending not at all in the fourth dimension, but up and
down, right and left, far and near.

We can see what should replace the light yellow face of
r0, when the section T\ comes in, by looking at the cube
60, fig. 107. What is distant in it one-quarter of an inch
from the light yellow face in the red direction ? It is an
ochre section with orange and pink lines and red points ;
see also fig. 103.

This square then forms the top square of Vi. Now we
can determine the nomenclature of all the regions of n by
considering what would be formed by the motion of this
square along a blue axis.

But we can adopt another plan. Let us take a hori-
zontal section of To, and finding that section in the figures,
of fig. 107 or fig. 103, from them determine what will
replace it, going on in the red direction.

A section of the ?'o cube has green, light blue, green,
light blue sides and blue points.

Now this square occurs on the base of each of the
section figures, 61, b2, etc. In them we see that £ inch in
the red direction from it lies a section witli brown and
light purple lines and purple corners, the interior being
of light brown. Hence this is the nomenclature of the
section which in ?'i replaces the section of To made from a
point along the blue axis.

Hence the colouring as given can be derived.

We have thus obtained a perfectly named group of
tesseracts. We can take a group of eighty-one of them
3x3x3x3, in four dimensions, and each tesseract will

have its name null, red, white, yellow, blue, vie, and

L88

THE FOURTH DIMENSION

whatever cubic view we take
what sides of the tesseracts
they touch each other.*

Tims, for instance, if we
si i own below, we can ask how

In the arrangement given
white, red, yellow, in space,
dimension. Hence we have
Imagine now the tesseractic
our space — we have first of

of them we can say exactly
we arc handling, and how

have the sixteen tesseracts
does null touch blue,
in fig. Ill we have the axes
blue running in the fourth

the ochre cubes as bases,
group to pass transverse to

all null ochre cube, white

i—

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\Och

re\

White

rs
ro

\ -

^x-Ligl

it brown

\ \

Red

Pink

Purple

Light

purple

Null

White

Blue

Light
blue

White

Ligl

it yello
A,

iv hiddt
bo

axis

Ligh

t green

direction

Fig. 111.

ochre cube, etc.; these instantly vanish, and we get the
section shown in the middle cube in fig. 103, and finally,
just when the tesseract block has moved one inch trans-
verse to our space, we have null ochre cube, and then
immediately afterwards the ochre cube of blue comes in.
Hence the tesseract null touches the tesseract blue by its
ochre cube, which is in contact, each and every point
of it, with the ochre cube of blue.

How does null touch white, we may ask? Looking at
the beginning A, fig. Ill, where we have the ochre

* At this point the reader will find it advantageous, if he has the
models, to go through the manipulations described in the appendix.

EEMAEKS on the figures

189

cubes, we see that null ochre touches white ochre by an
orange face. Now let us generate the null and white
tesseracts by a motion in the blue direction of each of
these cubes. Each of them generates the corresponding
tesseract, and the plane of contact of the cubes generates
the cube by which the tesseracts are in contact. Now an
orange plane carried along a blue axis generates a brown
cube. Hence null touches white by a brown cube.

If we ask again how red touches light blue tesseract,
let us rearrange our group, fig. 112, or rather turn it

v».5

White

axis

c-yo

White
hidden

c«yi

Fig. 112.

White
direction

Light yellow
hidden

about so that we have a different space view of it ; let
the red axis and the white axis run up and right, and let
the blue axis come in space towards us, then the yellow
axis runs in the fourth dimension. We have then two
blocks in which the bounding cubes of the tesseracts are
given, differently arranged with regard to us — the arrange-
ment is really the same, but it appears different to us.
Starting from the plane of the red and white axes we
have i lie four Bquares of the null, white, red, pink tesseracts
a- Bhown iii A, on the red, white plane, unaltered, only
from 1 Ik-im now comes out towards us the blue axis.

190 THE FOURTH DIMENSION

Hence we have null, white, red, pink tesseracts in contact
with our space by their cubes which have the red, white,
blue axis in them, that is by the light purple cubes.
Following on these four tesseracts we have that which
comes next to them in the blue direction, that is the
four blue, light blue, purple, light purple. These are
likewise in contact with our space by their light purple
cubes, so we see a block as named in the figure, of which
each cube is the one determined by the red, white, blue,
axes.

The yellow line now runs out of space ; accordingly one
inch on in the fourth dimension we come to the tesseracts
which follow on the eight named in C, fig. 112, in the
yellow direction.

These are shown in C.Vi, fig. 112. Between figure C
and C.Vi is that four-dimensional mass which is formed
by moving each of the cubes in C one inch in the fourth
dimension — that is, along a yellow axis ; for the yellow
axis now runs in the fourth dimension.

In the block C we observe that red (light purple
cube) touches light blue (light purple cube) by a point.
Now these two cubes moving together remain in contact
during the period in which they trace out the tesseracts
red and light blue. This motion is along the yellow
axis, consequently red and light blue touch by a yellow
line.

We have seen that the pink face moved in a yellow
direction traces out a cube ; moved in the blue direction it
also traces out a cube. Let us ask what the pink face
will trace out if it is moved in a direction within the
tesseract lying equally between the yellow and blue
directions. What section of the tesseract will it make ?

We will first consider the red line alone. Let us take
a cube with the red line in it and the yellow and blue
axes.

REMARKS ON THE FIGURES

191

Red

Yellow

The cube with the yellow, red, blue axes is shown in

fig. 113. If the red line is
moved equally in the yellow and
in the blue direction by four
equal motions of J inch each, it
takes the positions 11, 22, 33,
and ends as a red line.

Now, the whole of this red,
yellow, blue, or brown cube ap-
pears as a series of faces on the
successive sections of the tes-
seract starting from the ochre cube and letting the blue
axis run in the fourth dimension. Hence the plane
traced out by the red line appears as a series of lines in
the successive sections, in our ordinary way of representing
the tesseract; these lines are, in different places in each
successive section.

Blue
Fig. 113.

Null

YelloW

Null White

Fig. 114

Thus drawing our initial cube and the successive
sections, calling them /;0, bh 62, b3, b4, fig. 115, we have
the red line subject to this movement appearing in the
positions indicated.

We will now investigate what positions in the tesseract
another line in the pink face assumes when it is moved in
a similar manner.

Take a section of the original cube containing a vertical
line, 4, in the pink plane, fig. 115, We have, in the
section, the yellow direction, but not the blue.

192

'TIE FOrilTII DIMKXSION

From this section ;i cube goes off in the fourth dimen-
sion, which is formed by moving each point of the section
in the blue direction.

Yellow"

Null Whit?
Fig. 115.

Urrht blue White
Fig. 116.

Drawing this cube we have fig. 116.

Now this cube occurs as a series of sections in our
original representation of the tesseract. Taking four steps
as before this cube appears as the sections drawn in b0, bit
&2j ^3> ^4> fig- H7, and if the line 4 is subjected to a
movement equal in the blue and yellow directions, it will
occupy the positions designated by 4, 4lf 42, 43, 4i.

F\V\[\5\[\X

Fiff. 117.

Hence, reasoning in a similar manner about every line,
it is evident that, moved equally in the blue and yellow
directions, the pink plane will trace out a space which is
shown by the series of section planes represented in the
diagram.

Thus the space traced out by the pink face, if it is
moved equally in the yellow and blue directions, is repre-
sented by the set of planes delineated in Fig. 118, pink

REMARKS ON THE FIGURES

193

face or 0, then 1, 2, 3, and finally pink face or 4. This
solid is a diagonal solid of the tesseract, running from a
pink face to a pink face. Its length is the length of the
diagonal of a square, its side is a square.

Let us now consider the unlimited space which springs
from the pink face extended.

This space, if it goes off in the yellow direction, gives
us in it the ochre cube of the tesseract. Thus, if we have
the pink face given and a point in the ochre cube, we
have determined this particular space.

Similarly going off from the pink face in the blue
direction is another space, which gives us the light purple
cube of the tesseract in it. And any point being taken in

Pink

Null br

\ .\

Fi?. 118.

the light purple cube, this space going off from the pink
face is fixed.

The space we are speaking of can be conceived as
swinging round the pink face, and in each of its positions
it cuts out a solid figure from the tesseract, one of which
we have seen represented in fig. 118.

Each of these solid figures is given by one position of
the swinging space, and by one only. Hence in each of
them, if one point is taken, the particular one of the
slanting spaces is fixed. Thus we see that given a plane
and a point out of it a space is determined.

Now, two points determine a line.

Again, think of a line and a point outside it. Imagine
a plane rotating round the line. At some time in its
rotation it parses through the point. Thus a line, and a

194

THE FOURTH DIMENSION

point, or three points, determine a plane. And finally
four points determine a space. We have seen that a
plane and a point determine a space, and that three
points determine a plane ; so four points will determine
a space.

These four points may be any points, and we can take,
for instance, the four points at the extremities of the red,
white, yellow, blue axes, in the tesseract. These will
determine a space slanting with regard to the section
spaces we have been previously considering. This space
will cut the tesseract in a certain figure.

One of the simplest sections of a cube by a plane is
that in which the plane passes through the extremities
of the three edges which meet in a point. We see at
once that this plane would cut the cube in a triangle, but
we will go through the process by which a plane being
would most conveniently treat the problem of the deter-
mination of this shape, in order that we may apply the
method to the determination of the figure in which a
space cuts a tesseract when it passes through the 4
points at unit distance from a corner.

We know that two points determine a line, three points
determine a plane, and given any two points in a plane
the line between them lies wholly in the plane.

Let now the plane being study the section made by

a plane passing through the
null r, null wh, and null y
points, fig. 119. Looking at
the orange square, which, as
usual, we suppose to be ini-
tially in his plane, he sees
that the line from null r to
null y, which is a line in the
section plane, the plane, namely, through the three
extremities of the edges meeting in null, cuts the orange

Null

Null-wh.

REMARKS ON THE FIGURES 195

face in an orange line with null points. This then is one
of the boundaries of the section figure.

Let now the cube be so turned that the pink face
comes in his plane. The points null r and null wh
are now visible. The line between them is pink
with null points, and since this line is common to
the surface of the cube and the cutting plane, it is
a boundary of the figure in which the plane cuts the
cube.

Again, suppose the cube turned so that the light
yellow face is in contact with the plane being's plane.
He sees two points, the null ivh and the null y. The
line between these lies in the cutting plane. Hence,
since the three cutting lines meet and enclose a portion
of the cube between them, he has determined the
figure he sought. It is a triangle with orange, pink,
and light yellow sides, all equal, and enclosing an
ochre area.

Let us now determine in what figure the space,
determined by the four points, null r, null y, null
wh, null b, cuts the tesseract. We can see three
of these points in the primary position of the tesseract
resting against our solid sheet by the ochre cube.
These three points determine a plane which lies in
the space we are considering, and this plane cuts
the ochre cube in a triangle, the interior of which
Lfl ochre (fig. 119 will serve for this view), with pink,
light yellow and orange sides, and null points. Going
in the fourth direction, in one sense, from this plane
we pass into the tesseract, in the other sense we pass
away from it. The whole area inside the triangle is
common to the cutting plane we see, and a boundary
of the tesseract. Hence we conclude that the triangle
drawn is common to the tesseract and the cutting
space.

106

TIIH FOURTH DIMENSION

Now let the oclire cube turn out and the brown cube

come in. The dotted lines
show the position the ochre
cube has left (fig. 120).

Here we see three out
of the four points through
which the cutting plane
passes, null r, null y, and
null b. The plane they

or.

Sr- l^vNull-r. i

pur.

Red ;

?llow ^-^

Null.b.Blue Null
Fig. 120.

determine lies in the cutting space, and this plane
cuts out of the brown cube a triangle with orange,
purple and green sides, and null points. The orange
line of this figure is the same as the orange line in
the last figure.

Now let the light purple cube swing into our space,
towards us, fig. 121.

The cutting space which passes through the four points,

null r, y, wh, 6, passes through
the null r, wh, b, and there-
fore the plane these determine
lies in the cutting space.

This triangle lies before us.
It has a light purple interior
and pink, light blue, and
purple edges with null points.
This, since it is all of the
plane that is common to it, and this bounding of the
tesseract, gives us one of the bounding faces of our sec-
tional figure. The pink line in it is the same as the
pink line we found in the first figure — that of the ochre
cube.

Finally, let the tesseract swing about the light yellow
plane, so that the light green cube comes into our space.
It will point downwards.

The three points, n.y, n.ivh, n.b, are in the cutting

Nullb.

Fig. 121.

REMARKS ON THE FIGURES

19'

Null

space, and the triangle they determine is common to

the tesseract and the cut-
ting space. Hence this
boundary is a triangle hav-
ing a light yellow line,
which is the same as the
light yellow line of the first
figure, a light blue line and
a green line.

We have now traced the
cutting space between every
set of three that can be
made out of the four points

Null-wb.

Null-b.

Fig. 122.

in which it cuts the tesseract, and have got four faces
which all join on to each other by lines.

The triangles are shown in fig. 123 as they join on to
, nnr nllr the triangle in the ochre cube. But

they join on each to the other in an
exactly similar manner; their edges
are all identical two and two. They
form a closed figure, a tetrahedron,
enclosing a light brown portion which
is the portion of the cutting space
which lies inside the tesseract.

We cannot expect to see this light brown portion, any
more than a plane being could expect to see the inside
of a cube if an angle of it were pushed through his
plane. All he can do is to come upon the boundaries
of it in a different way to that in which he would if it
passed Btraight through his plane.

Thtlfl in this solid section; the whole interior lies per-
fectly open in the fourth dimension. Go round it as
we may we are simply looking at the boundaries of the
tesseracl which penetrates through our solid sheet. If
the tesseract were not to pass across so far, the triangle

198 THE FOURTH DIMENSION

would be smaller; if it were to pass farther, we should
have a different figure, the outlines of which can be
determined in a similar manner.

The preceding method is open to the objection that
it depends rather on our inferring what must be, than
our seeing what is. Let us therefore consider our
sectional space as consisting of a number of planes, each
very close to the last, and observe what is to be found
in each plane.

The corresponding method in the case of two dimen-
sions is as follows : — The plane
being can see that line of the
sectional plane through null y,
null w, null r, which lies in
Null-y.l4*~.-|— - -C\ the orange plane. Let him

now suppose the cube and the
Null Nullwh. . . , tip

p. 12. section plane to pass halt way

through his plane. Eeplacing

the red and yellow axes are lines parallel to them, sections

of the pink and light yellow faces.

Where will the section plane cut these parallels to
the red and yellow axes ?

Let him suppose the cube, in the position of the
drawing, fig. 124, turned so that the pink face lies
against his plane. He can see the line from the null r
point to the null wh point, and can see (compare fig. 119)
that it cuts ab a parallel to his red axis, drawn at a point
half way along the white line, in a point B, half way up.
I shall speak of the axis as having the length of an edge
of the cube. Similarly, by letting the cube turn so that
the light yellow square swings against his plane, he can
see (compare fig. 119) that a parallel to his yellow axis
drawn from a point half-way along the white axis, is cut
at half its length by the trace of the section plane in the
light yellow face.

REMARKS ON THE FIGURES 199

Hence when the cube had passed half-way through he
would have — instead of the orange line with null points,
which he had at first — an ochre line of half its length,
with pink and light yellow points. Thus, as the cube
passed slowly through his plane, he would have a suc-
cession of lines gradually diminishing in length and
forming an equilateral triangle. The whole interior would
be ochre, the line from which it started would be orange.
The succession of points at the ends of the succeeding
lines would form pink and light yellow lines and the
final point would be null. Thus looking at the successive
lines in the section plane as it and the cube passed across
his plane he would determine the figure cut out bit
by bit.

Coming now to the section of the tesseract, let us
imagine that the tesseract and its cutting space pass
slowly across our space ; we can examine portions of it,
and their relation to portions of the cutting space. Take
the section space which passes through the four points,
null r, ivh, y, b ; we can see in the ochre cube (fig. 119)
the plane belonging to this section space, which passes
through the three extremities of the red, white, yellow
axes.

Now let the tesseract pass half way through our space.
Instead of our original axes we have parallels to them,
purple, pink, and green, each of the same length as the
first axes, for the section of the tesseract is of exactly
the same shape as its ochre cube.

But the sectional space seen at this stage of the trans-
ference would not cut the section of the tesseract in a
plane disposed as at first.

To see where the sectional space would cut these
parallels to the original axes let the tesseract swing so
that, the orange face remaining stationary, the blue line
comes in to the left.

200

THE FOURTH DIMENSION

Green

'-^Yellow

Nullb. Blue Null
Fiff. 125.

Here (fig. 125) we have the null ?% y, b points, and of

the sectional space all we
see is the plane through these
three points in it.

In this figure we can draw
the parallels to the red and
yellow axes and see that, if
they started at a point half
way along the blue axis, they

would each be cut at a point so as to be half of their

previous length.

Swinging the tesseract into our space about the pink

face of the ochre cube we likewise find that the parallel

to the white axis is cut at half its length by the sectional

space.

Hence in a section made when the tesseract had passed

half across our space the parallels to the red, white, yellow

axes, which are now in our
space, are cut by the section
space, each of them half way
along, and for this stage of
the traversing motion we
should have fig. 126. The
section made of this cube by
the plane in which the sec-
tional space cuts it, is an

Blue L.blue bi.
Section b2 interior Light brown

Fig. 126.

equilateral triangle with purple, 1. blue, green points, and
1. purple, brown, 1. green lines.

Thus the original ochre triangle, with null points and
pink, orange, light yellow lines, would be succeeded by a
triangle coloured in manner just described.

This triangle would initially be only a very little smaller
than the original triangle, it would gradually diminish,
until it ended in a point, a null point. Each of its
edges would be of the same length. Thus the successive

EEMAEKS ON THE FIGURES

201

sections of the successive planes into which we analyse the
cutting space would be a tetrahedron of the description
shown (fig. 123), and the whole interior of the tetrahedron
would be light brown.

*c/7

Front view. The rear faces.

Fig. 127.

In fig. 127 the tetrahedron is represented by means of
its faces as two triangles which meet in the p. line, and
two rear triangles which join on to them, the diagonal
of the pink face being supposed to run vertically
upward.

We have now reached a natural termination. The
reader may pursue the subject in further detail, but will
find no essential novelty. I conclude with an indication
as to the manner in which figures previously given may
be used in determining sections by the method developed
above.

Applying this method to the tesseract, as represented
in Chapter IX., sections made by a space cutting the axes
equidistantly at any distance can be drawn, and also the
sections of tesseracts arranged in a block.

If we draw a plane, cutting all four axes at a point
six units distance from null, we have a slanting space.
This Bpace cuts the red, white, yellow axes in the

202

THE FOURTH DIMENSION

Red

points lmn (fig. 128), and so in the region of our space

before we go off into
the fourth dimension,
we have the plane
represented by lmn
extended. This is what
is common to the
slanting space and our
space.

This plane cuts the

low axis

White axis

Fig. 128.

ochre cube in the triangle efg.

Comparing this with (fig. 72) oh, we see that the
hexagon there drawn is part of the triangle efg.

Let us now imagine the tesseract and the slanting
space both together to pass transverse to our space, a
distance of one unit, we have in 1A a section of the
tesseract, whose axes are parallels to the previous axes.
The slanting space cuts them at a distance of five units
along each. Drawing the plane through these points in
\h it will be found to cut the cubical section of the
tesseract in the hexagonal figure drawn. In 2h (fig. 72) the
slanting space cuts the parallels to the axes at a distance
of four along each, and the hexagonal figure is the section
of this section of the tesseract by it. Finally when 3/t
comes in the slanting space cuts the axes at a distance
of three along each, and the section is a triangle, of which
the hexagon drawn is a truncated portion. After this
the tesseract, which extends only three units in each of
the four dimensions, has completely passed transverse
of our space, and there is no more of it to be cut. Hence,
putting the plane sections together in the right relations,
we have the section determined by the particular slanting
space : namely an octahedron.

•

CHAPTER XIV.*

A RECAPITULATION AND EXTENSION OF
THE PHYSICAL ARGUMENT

There are two directions of inquiry in which the
research for the physical reality of a fourth dimension
can be prosecuted. One is the investigation of the
infinitely great, the other is the investigation of the
infinitely small.

By the measurement of the angles of vast triangles,
whose sides are the distances between the stars, astronomers
have sought to determine if there is any deviation from
the values given by geometrical deduction. If the angles
of a celestial triangle do not together equal two right
angles, there would be an evidence for the physical reality
of a fourth dimension.

This conclusion deserves a word of explanation. If
space is really four-dimensional, certain conclusions follow
which must be brought clearly into evidence if we are to
frame the questions definitely which we put to Nature.
To account for our limitation let us assume a solid material
sheet against which we move. This sheet must stretch
alongside every object in every direction in which it

* The contents of thifl chapter are taken from a paper read before
the Philosophical Society <>f Washington. The mathematical portion
of the paper has appeared in part in the Transactions of the Royal
Irian Academy under the title, "Cayley'a formula' of orthogonal

transformation."

203

204 THE FOURTH DIMENSION

visibly moves. Every material body must slip or slide
along this sheet, not deviating from contact with it in
any motion which we can observe.

The necessity for this assumption is clearly apparent, if
we consider the analogous case of a suppositionary plane
world. If there were any creatures whose experiences
were confined to a plane, we must account for their
limitation. If they were free to move in every space
direction, they would have a three-dimensional motion ;
hence they must be physically limited, and the only way
in w^hich we can conceive such a limitation to exist is by
means of a material surface against which they slide.
The existence of this surface could onlv be known to
them indirectly. It does not lie in any direction from
them in which the kinds of motion they know of leads
them. If it were perfectly smooth and always in contact
with every material object, there would be no difference in
their relations to it which would direct their attention to it.

But if this surface were curved — if it were, say, in the
form of a vast sphere — the triangles they drew would
really be triangles of a sphere, and when these triangles
are large enough the angles diverge from the magnitudes
they would have for the same lengths of sides if the
surface were plane. Hence by the measurement of
triangles of very great magnitude a plane being might
detect a difference from the laws of a plane world in his
physical world, and so be led to the conclusion that there
was in reality another dimension to space — a third
dimension — as well as the two which his ordinary experi-
ence made him familiar with.

Now, astronomers have thought it worth while to
examine the measurements of vast triangles drawn from
one celestial body to another with a view to determine if
there is anything like a curvature in our space — that is to
say, they have tried astronomical measurements to find

RECAPITULATION AND EXTENSION 205

out if the vast solid sheet against which, on the sup-
position of a fourth dimension, everything slides is
curved or not. These results have been negative. The
solid sheet, if it exists, is not curved or, being curved, has
not a sufficient curvature to cause any observable deviation
from the theoretical value of the angles calculated.

Hence the examination of the infinitely great leads to
no decisive criterion. It neither proves nor disproves the
existence of a fourth dimension.

Coming now to the prosecution of the inquiry in the
direction of the infinitely small, we have to state the
question thus : Our laws of movement are derived from
the examination of bodies which move in three-dimensional
space. All our conceptions are founded on the sup-
position of a space which is represented analytically by
three independent axes and variations along them — that
is, it is a space in which there are three independent
movements. x\ny motion possible in it can be compounded
out of these three movements, which we may call : up,
right, away.

To examine the actions of the very small portions of
matter with the view of ascertaining if there is any
evidence in the phenomena for the supposition of a fourth
dimension of space, we must commence by clearly defining
what the laws of mechanics would be on the supposition
of a fourth dimension. It is of no use asking if the
phenomena of the smallest particles of matter are like —
we do not know what. We must have a definite con-
ception of what the laws of motion would be on the
supposition of the fourth dimension, and then inquire if
the phenomena of the activity of the smaller particles of
matter resemble the conceptions which we have elaborated.

Now, the task of forming these conceptions is by no
means one to be lightly dismissed. .Movement in space
has many feature? which differ entirely from movement

200 THE FOURTH DIMENSION

on a plane; and when wo sot, about to form the con-
ception of motion in four dimensions, we find that there
is at least as great a step as from the plane to three-
dimensional space.

I do not say that the step is difficult, but I want to
point out that it must be taken. When we have formed
the conception of four-dimensional motion, we can ask a
rational question of Nature. Before we have elaborated
our conceptions we are asking if an unknown is like an
unknown — a futile inquiry.

As a matter of fact, four-dimensional movements are in
every way simple and more easy to calculate than three-
dimensional movements, for four-dimensional movements
are simply two sets of plane movements put together.

Without the formation of an experience of four-
dimensional bodies, their shapes and motions, the subject
can be but formal — logically conclusive, not intuitively
evident. It is to this logical apprehension that I must
appeal.

It is perfectly simple to form an experiential familiarity
with the facts of four-dimensional movement. The
method is analogous to that which a plane being would
have to adopt to form an experiential familiarity with
three-dimensional movements, and may be briefly
summed up as the formation of a compound sense by
means of which duration is regarded as equivalent to
extension.

Consider a being confined to a plane. A square enclosed
by four lines will be to him a solid, the interior of which
can only by examined by breaking through the lines.
If such a square were to pass transverse to his plane, it
would immediately disappear. It would vanish, going in
no direction to which he could point.

If, now, a cube be placed in contact with his plane, its
surface of contact would appear like the square which we

RECAPITULATION AND EXTENSION 207

have just mentioned. But if it were to pass transverse to
his plane, breaking through it, it would appear as a lasting
square. The three-dimensional matter will give a lasting
appearance in circumstances under which two-dimensional
matter will at once disappear.

Similarly, a four-dimensional cube, or, as we may call
it, a tesseract, which is generated from a cube by a
movement of every part of the cube in a fourth direction
at right angles to each of the three visible directions in
the cube, if it moved transverse to our space, would
appear as a lasting cube.

A cube of three-dimensional matter, since it extends to
no distance at all in the fourth dimension, would instantly
disappear, if subjected to a motion transverse to our space.
It would disappear and be gone, without it being possible
to point to any direction in which it had moved.

All attempts to visualise a fourth dimension are futile. It
must be connected with a time experience in three space.
The most difficult notion for a plane being to acquire
would be that of rotation about a line. Consider a plane
being facing a square. If he were told that rotation
about a line were possible, he would move his square this
way and that. A square in a plane can rotate about a
point, but to rotate about a line would seem to the plane
being perfectly impossible. How could those parts of his
square which were on one side of an edge come to the
other side without the edge moving? He could under-
stand their reflection in the edge. He could form an
idea of the looking-glass image of his square lying on the
opposite side of the line of an edge, but by no motion
that he knows of can he make the actual square assume
thai position. The result of the rotation would be like
reflection in the edge, but it would be a physical im-
possibility to produce it in the plane.
The demonstration of rotation about a line must be to

'Jns THE FOURTH DIMENSION

him purely formal. If he conceived the notion of a cube
stretching out in an unknown direction away from his
plane, then he can see the base of it, his square in the
plane, rotating round a point. He can likewise apprehend
that every parallel section taken at successive intervals in
the unknown direction rotates in like manner round a
point. Thus he would come to conclude that the whole
body rotates round a line — the line consisting of the
succession of points round which the plane sections rotate.
Thus, given three axes, x, y, z, if x rotates to take the
the place of y, and y turns so as to point to negative x,
then the third axis remaining unaffected by this turning
is the axis about which the rotation takes place. This,
then, would have to be his criterion of the axis of a
rotation — that which remains unchanged when a rotation
of every plane section of a body takes place.

There is another way in which a plane being can think
about three-dimensional movements ; and, as it affords
the type by which we can most conveniently think about
four-dimensional movements, it will be no loss of time to
consider it in detail.

We can represent the plane being and his object by
figures cut out of paper, which slip on a smooth surface.
The thickness of these bodies must be taken as so minute
y that their extension in the third di-

mension escapes the observation of the
plane being, and he thinks about them
as if they were mathematical plane
\B' figures in a plane instead of being

material bodies capable of moving on

A 8 * a plane surface. Let ax, ky be two

Fig. l (129). axes and abcd a square. As far as

movements in the plane are concerned, the square can

rotate about a point A, for example. It cannot rotate

about a side, such as AC.

RECAPITULATION AND EXTENSION 209

But if the plane being is aware of the existence of a
third dimension he can study the movements possible in
the ample space, taking his figure portion by portion.

His plane can only hold two axes. But, since it can
hold two, he is able to represent a turning into the third
dimension if he neglect one of his axes and represent the
third axis as lying in his plane. He can make a drawing
in his plane of what stands up perpendicularly from his

plane. Let az be the axis, which
stands perpendicular to his plane at
A. He can draw in his plane two
# lines to represent the two axes, ax
and kz. Let Fig. 2 be this draw-
ing. Here the z axis has taken

B the place of the y axis, and the

Fig. 2 (130). plane of ax az is represented in his

plane. In this figure all that exists of the square abcd
will be the line ab.

The square extends from this line in the y direction,
but more of that direction is represented in Fig. 2. The
plane being can study the turning of the line ab in this
diagram. It is simply a case of plane turning around the
point A. The line ab occupies intermediate portions like abx
and after half a revolution will lie on Ax produced through a.

Now, in the same way, the plane being can take
another point, a', and another line, a'b', in his square.
He can make the drawing of the two directions at a', one
along a'b', the other perpendicular to his plane. He
will obtain a figure precisely similar to Fig. 2, and will
see that, as AB can turn around A, so a'c' around a.

In tli is turning ab and a'b' would not interfere with
each other, as they would if they moved in the plane
around the separate points A and a'.

Hence the plane being would conclude that a rotation
round a line was possible. He could see his square as it

210 THE FOURTH DIMENSION

began to make this turning. He could see it half way
round when it came to lie on the opposite side of the line
AC. But in intermediate portions he could not see it,
for it runs out of the plane.

Coming now to the question of a four-dimensional body,
let us conceive of it as a series of cubic sections, the first
in our space, the rest at intervals, stretching away from
our space in the unknown direction.

We must not think of a four-dimensional body as
formed by moving a three-dimensional body in any
direction which we can see.

Eefer for a moment to Fig. 3. The point A, moving to
the right, traces out the line AC. The line AC, moving
away in a new direction, traces out the square aceg at
the base of the cube. The square AEGC, moving in a
new direction, will trace out the cube acegbdhf. The
vertical direction of this last motion is not identical with
any motion possible in the plane of the base of the cube.
It is an entirely new direction, at right angles to every
line that can be drawn in the base. To trace out a
tesseract the cube must move in a new direction — a
direction at right angles to any and every line that can
be drawn in the space of the cube.

The cubic sections of the tesseract are related to the
cube we see, as the square sections of the cube are related
to the square of its base which a plane being sees.

Let us imagine the cube in our space, which is the base
of a tesseract, to turn about one of its edges. The rotation
will carry the whole body with it, and each of the cubic
sections will rotate. The axis we see in our space will
remain unchanged, and likewise the series of axes parallel
to it about which each of the parallel cubic sections
rotates. The assemblage of all of these is a plane.

Hence in four dimensions a body rotates about a plane.
There is no such thing as rotation round an axis.

[RECAPITULATION AND EXTENSION 211

We may regard the rotation from a different point of
view. Consider four independent axes each at right
angles to all the others, drawn in a four-dimensional body.
Of these four axes we can see any three. The fourth
extends normal to our space.

Rotation is the turning of one axis into a second, and
the second turning to take the place of the negative of
the first. It involves two axes. Thus, in this rotation of
a four-dimensional body, two axes change and two remain
at rest. Four-dimensional rotation is therefore a turning
about a plane.

As in the case of a plane being, the result of rotation
about a line would appear as the production of a looking-
glass image of the original object on the other side of the
line, so to us the result of a four-dimensional rotation
would appear like the production of a looking-glass image
of a body on the other side of a plane. The plane would
be the axis of the rotation, and the path of the body
between its two appearances would be unimaginable in
three-dimensional space.

Let us now apply the method by which a plane being

could examine the nature of rota-
tion about a line in our examination
of rotation about a plane. Fig. 3
represents a cube in our space, the
three axes x, y, z denoting its
three dimensions. Let iv represent
1 he fourth dimension. Now, since
A C * in our space we can represent any

Fig. 3 (131). three dimensions, we can, if we

choose, make a representation of what is in the .-pare
determined by the three axes x, z, w. This is a three-
dimensional space determined by two of the axes we have
drawn, ./• and ;. and in place of y lh<1 fourth axis, w. We
cannot, keeping x and z, have both y and w in our space ;

THE FOURTH DIMENSION

so we will let y go and draw w in its place. What will be
our view of the cube ?

Evidently we shall have simply the square that is in

the plane of xz, the square acdb.

The rest of the cube stretches in

the y direction, and, as we have

none of the space so determined,

we have only the face of the cube.

This is represented in fig. 4.

A c Now, suppose the whole cube to

Fig. 4 (132). kg turned from the x to the w

direction. Conformably with our method, we will not

take the whole of the cube into consideration at once, but

will begin with the face A BCD.

Let this face begin to turn. Fig. 5

represents one of the positions it will

occupy ; the line ab remains on the

z axis. The rest of the face extends

between the x and the w direction.

Now, since we can take any three

A x axes, let us look at what lies in

Fig. 5 (133). the space of zyiv, and examine the

turning there. We must now let the z axis disappear

and let the w axis run in the direction in which the z ran.

Making this representation, what

do we see of the cube ? Obviously

we see only the lower face. The rest

of the cube lies in the space of xyz.

In the space of xyz we have merely

A c the base of the cube lying in the

Fig. G (134). plane of xy, as shown in fig. 6.

Now let the x to iv turning take place. The square

aceg will turn about the line ae. This edge will

remain along the y axis and will be stationary, however

far the square turns.

[RECAPITULATION AND EXTENSION

213

Thus, if the cube be turned by an x to w turning, both
the edge ab and the edge AC remain
stationary ; hence the whole face
abef in the yz plane remains fixed.
The turning has taken place about
the face abef.

Suppose this turning to continue

^ * till AC runs to the left from A.

Fig. 7 (13d). The cube will occupy the position

shown in fig. 8. This is the looking-glass image of the

cube in fig. 3. By no rotation in three-dimensional space

can the cube be brought from
the position in fig. 3 to that
shown in fig. 8.

We can think of this turning
as a turning of the face abcd
about AB, and a turning of each
section parallel to abcd round
the vertical line in which it
intersects the face abef, the
space in which the turning takes place being a different
one from that in which the cube lies.

One of the conditions, then, of our inquiry in the
direction of the infinitely small is that we form the con-
ception of a rotation about a plane. The production of a
body in a state in which it presents the appearance of a
looking-glass image of its former state is the criterion
for a four-dimensional rotation.

There is some evidence for the occurrence of such trans-
formations of bodies in the change of bodies from those
which produce a right-handed polarisation of light to
those which produce a left-handed polarisation; but this
La not a point to which any very great importance can
be attached.

Still, in this connection, let me quote a remark from

2"? position Imposition

Fig. 8 (136).

214 THE FOURTH DIMENSION

Prof. John G. McKendrick's address on Physiology before
the British Association at Glasgow. Discussing the
possibility of the hereditary production of characteristics
through the material structure of the ovum, he estimates
that in it there exist 12,000,000,000 biophors, or ultimate
particles of living matter, a sufficient number to account
for hereditary transmission, and observes : " Thus it is
conceivable that vital activities may also be determined
by the kind of motion that takes place in the molecules
of that which we speak of as living matter. It may be
different in kind from some of the motions known to
physicists, and it is conceivable that life may be the
transmission to dead matter, the molecules of which have
already a special kind of motion, of a form of motion
sui generis."

Now, in the realm of organic beings symmetrical struc-
tures— those with a right and left symmetry — are every-
where in evidence. Granted that four dimensions exist,
the simplest turning produces the image form, and by a
folding-over structures could be produced, duplicated
right and left, just as is the case of symmetry in a
plane.

Thus one very general characteristic of the forms of
organisms could be accounted for by the supposition that
a four-dimensional motion was involved in the process of
life.

But whether four-dimensional motions correspond in
other respects to the physiologist's demand for a special
kind of motion, or not, I do not know. Our business is
with the evidence for their existence in physics. For
this purpose it is necessary to examine into the signifi-
cance of rotation round a plane in the case of extensible
and of fluid matter.

Let us dwell a moment longer on the rotation of a rigid
body. Looking at the cube in fig. 3, which turns about

RECAPITULATION AND EXTENSION

215

the face of abfe, we see that any line in the face can
take the place of the vertical and horizontal lines we have
examined. Take the diagonal line af and the section
through it to gh. The portions of matter which were on
one side of af in this section in fig. 3 are on the
opposite side of it in fig. 8. They have gone round the
line af. Thus the rotation round a face can be considered
as a number of rotations of sections round parallel lines
in it.

The turning about two different lines is impossible in
three-dimensional space. To take another illustration,
suppose A and B are two parallel lines in the xy plane,
and let CD and ef be two rods crossing them. Now, in
the space of xyz if the rods turn round the lines A and B

in the same direction they
will make two independent
circles.

When the end f is going
down the end c will be coming
up. They will meet and con-
x flict.

But if we rotate the rods
about the plane of ab by the
z to iv rotation these move-
ments will not conflict. Sup-
pose all the figure removed
with the exception of the plane xz, and from this plane
draw the axis of w, so that we are looking at the space
of xziv.

Here, fig. 10, we cannot see the lines A and B. We
see the points g and n, in which A and B intercept
the x axis, but we cannot see the lines themselves, for
they run in the y direction, and that is not in our
drawing.

Now, if the rods move with the z to w rotation they will

Fig. 9 (137).

210 THE FOURTH DIMENSION

turn in parallel planes, keeping their relative positions.

The point i>, for instance, will
describe a circle. At one time
it will be above the line A, at
another time below it. Hence
it rotates round A.

Not only two rods but any
number of rods crossing the
plane wTill move round it har-
moniously. We can think of

„. n „„„x this rotation by supposing the

Fig. 10(138). r r

rods standing up irom one line

to move round that line and remembering that it is

not inconsistent with this rotation for the rods standing

up along another line also to move round it, the relative

positions of all the rods being preserved. Now, if the

rods are thick together, they may represent a disk of

matter, and we see that a disk of matter can rotate

round a central plane.

Kotation round a plane is exactly analogous to rotation
round an axis in three dimensions. If we want a rod to
turn round, the ends must be free ; so if we want a disk
of matter to turn round its central plane by a four-dimen-
sional turning, all the contour must be free. The whole
contour corresponds to the ends of the rod. Each point
of the contour can be looked on as the extremity of an
axis in the body, round each point of wThich there is a
rotation of the matter in the disk.

If the one end of a rod be clamped, we can twist the
rod, but not turn it round ; so if any part of the contour
of a disk is clamped we can impart a twist to the disk,
but not turn it round its central plane. In the case of
extensible materials a long, thin rod will twist round its
axis, even when the axis is curved, as, for instance, in the
case of a ring of India rubber.

RECAPITULATION AND EXTENSION

217

Fig. 11 (139).

In an analogous manner, in four dimensions we can have
rotation round a curved plane, if I may use the expression.
A sphere can be turned inside out in four dimensions.

Let fig. 11 represent a
spherical surface, on each
side of which a layer of
matter exists. The thick-
ness of the matter is rep-
resented by the rods CD and
ef, extending equally with-
out and within.

Now, take the section of
the sphere by the yz plane
we have a circle — fig. 12.
Now, let the w axis be drawn
in place of the x axis so that
we have the space of yzw
represented. In this space all that there will be seen of
the sphere is the circle drawn.

Here we see that there is no obstacle to prevent the

rods turning round. If
the matter is so elastic
that it will give enough
for the particles at E and
c to be separated as they
are at F and D, they
can rotate round to the
position D and F, and a
similar motion is possible
for all other particles.
There is no matter" or
obstacle to prevent them
1 ''-■ ,J ' ' 10)- from moving out in the

W direction] and then on round the circumference as an
axis. Now, what will hold for one section will hold for

218 THE FOURTH DIMENSION

all, as the fourth dimension is at right angles to all the
sections which can be made of the sphere.

We have supposed the matter of which the sphere is
composed to be three-dimensional. If the matter had a
small thickness in the fourth dimension, there would be
a slight thickness in fig. 12 above the plane of the paper
— a thickness equal to the thickness of the matter in the
fourth dimension. The rods would have to be replaced
by thin slabs. But this would make no difference as to
the possibility of the rotation. This motion is discussed
by Newcomb in the first volume of the American Journal
of Mathematics.

Let us now consider, not a merely extensible body, but
a liquid one. A mass of rotating liquid, a whirl, eddy,
or vortex, has many remarkable properties. On first
consideration we should expect the rotating mass of
liquid immediately to spread off and lose itself in the
surrounding liquid. The water flies off a wheel whirled
round, and we should expect the rotating liquid to be
dispersed. But see the eddies in a river strangely per-
sistent. The rings that occur in puffs of smoke and last
so long are whirls or vortices curved round so that their
opposite ends join together. A cyclone will travel over
great distances.

Helmholtz was the first to investigate the properties of
vortices. He studied them as they would occur in a perfect
fluid — that is, one without friction of one moving portion
or another. In such a medium vortices would be inde-
structible. They would go on for ever, altering their
shape, but consisting always of the same portion of the
fluid. But a straight vortex could not exist surrounded
entirely by the fluid. The ends of a vortex must reach to
some boundary inside or outside the fluid.

A vortex which is bent round so that its opposite ends
join is capable of existing, but no vortex has a free end in

RECAPITULATION AND EXTENSION 219

the fluid. The fluid round the vortex is always in motion,
and one produces a definite movement in another.

Lord Kelvin has proposed the hypothesis that portions
of a fluid segregated in vortices account for the origin of
matter. The properties of the ether in respect of its
capacity of propagating disturbances can be explained
by the assumption of vortices in it instead of by a pro-
perty of rigidity. It is difficult to conceive, however,
of any arrangement of the vortex rings and endless vortex
filaments in the ether.

Now, the further consideration of four-dimensional
rotations shows the existence of a kind of vortex which
would make an ether filled with a homogeneous vortex
motion easily thinkable.

To understand the nature of this vortex, we must go
on and take a step by which we accept the full signifi-
cance of the four-dimensional hypothesis. Granted four-
dimensional axes, we have seen that a rotation of one into
another leaves two unaltered, and these two form the
axial plane about which the rotation takes place. But
what about these two ? Do they necessarily remain
motionless ? There is nothing to prevent a rotation of
these two, one into the other, taking place concurrently
with the first rotation. This possibility of a double
rotation deserves the most careful attention, for it is the
kind of movement which is distinctly typical of four
dimensions.

Rotation round a plane is analogous to rotation round
an axis. But in three-dimensional space there is no
motion analogous to the double rotation, in which, while
axis 1 changes into axis 2, axis 3 changes into axis 4.

< lonsider a four-dimensional body, with four independent

. .'", y, z, w. A point in it can move in only one

direction at a given moment, tf tli<v body has a velocity

of rotation by which the x axis changes into the y axis

220 THE FOURTH DIMENSION

and all parallel sections move in a similar manner, then
the point will describe a circle. If, now, in addition to
the rotation by which the x axis changes into the y axis the
body has a rotation by which the z axis turns into the
w axis, the point in question will have a double motion
in consequence of the two turnings. The motions will
compound, and the point will describe a circle, but not
the same circle which it would describe in virtue of either
rotation separately.

We know that if a body in three-dimensional space is
given two movements of rotation they will combine into a
single movement of rotation round a definite axis. It is
in no different condition from that in which it is sub-
jected to one movement of rotation. The direction of
the axis changes ; that is all. The same is not true about
a four-dimensional body. The two rotations, x to y and
z to w, are independent. A body subject to the two is in
a totally different condition to that which it is in when
subject to one only. When subject to a rotation such as
that of x to y, a whole plane in the body, as we have
seen, is stationary. When subject to the double rotation
no part of the body is stationary except the point common
to the two planes of rotation.

If the two rotations are equal in velocity, every point
in the body describes a circle. All points equally distant
from the stationary point describe circles of equal size.

We can represent a four-dimensional sphere by means
of two diagrams, in one of which we take the three axes,
x, y, z ; in the other the axes x, w, and z. In fig. 13 we
have the view of a four-dimensional sphere in the space of
xyz. Fig. 13 shows all that we can see of the four
sphere in the space of xyz, for it represents all the
points in that space, which are at an equal distance from
the centre.

Let us now take the xz section, and let the axis of iv

RECAPITULATION AND EXTENSION

991

take the place of the y axis. Here, in fig. 14, we have
the space of xzw. In this space we have to take all the
points which are at the same distance from the centre,
consequently we have another sphere. If we had a three-
dimensional sphere, as has been shown before, we should
have merely a circle in the xzw space, the xz circle seen
in the space of xzw. But now, taking the view in the
space of xzw. we have a sphere in that space also. In a
similar manner, whichever set of three axes we take, we
obtain a sphere.

p' Showing axes
xyz

Showing axes
xwz

Fig. 13 (141).

Fig. 14 (142).

In fig. 13, let us imagine the rotation in the direction
xy to be taking place. The point x will turn to y, and p
to jj. The axis zz remains stationary, and this axis is all
of the plane ztu which we can see in the space section
exhibited in the figure.

In fig. 14, imagine the rotation from z to w to be taking
place. The w axis now occupies the position previously
occupied by the y axis. This does not mean that the
to axis can coincide with the y axis. It indicates that we
are looking at the four-dimensional sphere from a different
point of view. Any three-space view will show us three
axes, and in fig. 14 we are looking at xzw.

The only part that is identical in the two diagrams is
the circle of the x and z axes, which axes are contains 1
in both diagrams. Thus the plane zxz is the same in
both, and the point jj represents the same point in both

222 THE FOURTH DIMENSION

diagrams. Now, in fig. 14 let the zw rotation take place,
the z axis will turn toward the point w of the w axis, and
the point p will move in a circle about the point x.

Thus in fig. 13 the point p moves in a circle parallel to
the xy plane ; in fig. 14 it moves in a circle parallel to the
zw plane, indicated by the arrow.

Now, suppose both of these independent rotations com-
pounded, the point p will move in a circle, but this circle
will coincide with neither of the circles in which either
one of the rotations will take it. The circle the point p
will move in will depend on its position on the surface of
the four sphere.

In this double rotation, possible in four-dimensional
space, there is a kind of movement totally unlike any
with which we are familiar in three-dimensional space.
It is a requisite preliminary to the discussion of the
behaviour of the small particles of matter, with a view to
determining whether they show the characteristics of four-
dimensional movements, to become familiar with the main
characteristics of this double rotation. And here I must
rely on a formal and logical assent rather than on the
intuitive apprehension, which can only be obtained by a
more detailed study.

In the first place this double rotation consists in two
varieties or kinds, which we will call the A and B kinds.
Consider four axes, x, y, z, w. The rotation of x to y can
be accompanied with the rotation of z to w. Call this
the A kind.

But also the rotation of x to y can be accompanied by
the rotation, of not z to iv, but w to z. Call this the
B kind.

They differ in only one of the component rotations. One
is not the negative of the other. It is the semi-negative.
The opposite of an x to y, z to w rotation would be y to x}
w to z. The semi-negative is x to y and w to z.

RECAPITULATION AND EXTENSION 223

If four dimensions exist and we cannot perceive them,
because the extension of matter is so small in the fourth
dimension that all movements are withheld from direct
observation except those which are three-dimensional, we
should not observe these double rotations, but only the
effects of them in three-dimensional movements of the
type with which we are familiar.

If matter in its small particles is four-dimensional,
we should expect this double rotation to be a universal
characteristic of the atoms and molecules, for no portion
of matter is at rest. The consequences of this corpus-
cular motion can be perceived, but only under the form
of ordinary rotation or displacement. Thus, if the theory
of four dimensions is true, we have in the corpuscles of
matter a whole world of movement, which we can never
study directly, but only by means of inference.

The rotation A, as I have defined it, consists of two
equal rotations — one about the plane of ziv, the other
about the plane of xy. It is evident that these rotations
are not necessarily equal. A body may be moving with a
double rotation, in which these two independent com-
ponents are not equal ; but in such a case we can consider
the body to be moving with a composite rotation — a
rotation of the A or B kind and, in addition, a rotation
about a plane.

If we combine an A and a B movement, we obtain a
rotation about a plane; for, the first being x to y and
z to iv, and the second being x to y and to to z, when they
are put together the z to w and w to z rotations neutral ise
each other, and we obtain an x to y rotation only, which
is a rotation about the plane of zw. Similarly, if we
take a B rotation, y to x and z to w, we get, on combining
this with the A rotation, a rotation of z to w about the
xy plane. In this case the plane of rotation is in the
three-dimensional Bpace of xyz, and we have what has

22 1 THE FOURTH DIMENSION

been described before— a twisting about a plane in our
space.

Consider now a portion of a perfect liquid having an A
motion. It can be proved that it possesses the properties
of a vortex. It forms a permanent individuality — a
separated-out portion of the liquid — accompanied by a
motion of the surrounding liquid. It has properties
analogous to those of a vortex filament. But it is not
necessary for its existence that its ends should reach the
boundary of the liquid. It is self-contained and, unless
disturbed, is circular in every section.

If we suppose the ether to have its properties of trans-
mitting vibration given it by such vortices, we must
inquire how they lie together in four-dimensional space.
Placing a circular disk on a plane and surrounding it by
six others, we find that if the central one is given a motion
of rotation, it imparts to the others a rotation which is

antagonistic in every two ad-
jacent ones. If A goes round,
as shown by the arrow, B and
c will be moving in opposite
ways, and each tends to de-
stroy the motion of the other.
Now, if we suppose spheres
to be arranged in a corre-
sponding manner in three-
dimensional space, they will
Fig. 15(143). , a '4R \- i

be grouped in figures which

are for three-dimensional space what hexagons are for
plane space. If a number of spheres of soft clay be
pressed together, so as to fill up the interstices, each will
assume the form of a fourteen-sided figure called a
te t rakaidecagon .

Now, assuming space to be filled with such tetrakai-
decagons, and placing a sphere in each, it will be found

[RECAPITULATION AND EXTENSION 225

that one sphere is touched by six others. The remaining
eight spheres of the fourteen which surround the central
one will not touch it, but will touch three of those in
contact with it. Hence, if the central sphere rotates, it
will not necessarily drive those around it so that their
motions will be antagonistic to each other, but the
velocities will not arrange themselves in a systematic
manner.

In four-dimensional space the figure which forms the
next term of the series hexagon, tetrakaidecagon, is a
thirty-sided figure. It has for its faces ten solid tetra-
kaidecagons and twenty hexagonal prisms. Such figures
will exactly fill four-dimensional space, five of them meet-
ing at every point. If, now, in each of these figures we
suppose a solid four-dimensional sphere to be placed, any
one sphere is surrounded by thirty others. Of these it
touches ten, and, if it rotates, it drives the rest by means
of these. Now, if we imagine the central sphere to be
given an A or a B rotation, it will turn the whole mass of
sphere round in a systematic manner. Suppose four-
dimensional space to be filled with such spheres, each
rotating with a double rotation, the whole mass would
form one consistent system of motion, in which each one
drove every other one, with no friction or lagging behind.

Every sphere would have the same kind of rotation. In
three-dimensional space, if one body drives another round
the second body rotates with the opposite kind of rotation ;
Inn in four-dimensional space these four-dimensional
spheres would each have the double negative of the rotation
of the one next it, and we have seen that the double
negative of an A or B rotation is still an A or B rotation.
Thus four-dimensional space could be filled with a system
of Belf-preservative living energy. If we imagine the
fonr-ditnensiona] spheres to be of liquid and not of solid
matter, then, even if the liquid were not quite perfect and

226 THE FOURTH DIMENSION

there were a slight retarding effect of one vortex on
another, the system would still maintain itself.

In this hypothesis we must look on the ether as
possessing energy, and its transmission of vibrations, not
as the conveying of a motion imparted from without, but
as a modification of its own motion.

We are now in possession of some of the conceptions of
four-dimensional mechanics, and will turn aside from the
line of their development to inquire if there is any
evidence of their applicability to the processes of nature.

Is there any mode of motion in the region of the
minute which, giving three-dimensional movements for
its effect, still in itself escapes the grasp of our mechanical
theories ? I would point to electricity. Through the
labours of Faraday and Maxwell we are convinced that the
phenomena of electricity are of the nature of the stress
and strain of a medium ; but there is still a gap to be
bridged over in their explanation — the laws of elasticity,
which Maxwell assumes, are not those of ordinary matter.
And, to take another instance : a magnetic pole in the
neighbourhood of a current tends to move. Maxwell has
shown that the pressures on it are analogous to the
velocities in a liquid which would exist if a vortex took
the place of the electric current ; but we cannot point out
the definite mechanical explanation of these pressures.
There must be some mode of motion of a body or of the
medium in virtue of which a body is said to be
electrified.

Take the ions which convey charges of electricity 500
times greater in proportion to their mass than are carried
by the molecules of hydrogen in electrolysis. In respect
of what motion can these ions be said to be electrified ?
It can be shown that the energy they possess is not
energy of rotation. Think of a short rod rotating. If it
is turned over it is found to be rotating in the opposite

RECAPITULATION AND EXTENSION 227

direction. Now, if rotation in one direction corresponds to
positive electricity, rotation in the opposite direction cor-
responds to negative electricity, and the smallest electrified
particles would have their charges reversed by being
turned over — an absurd supposition.

If we fix on a mode of motion as a definition of
electricity, we must have two varieties of it, one for
positive and one for negative ; and a body possessing the
one kind must not become possessed of the other by any
change in its position.

All three-dimensional motions are compounded of rota-
tions and translations, and none of them satisfy this first
condition for serving as a definition of electricity.

But consider the double rotation of the A and B kinds.
A body rotating with the A motion cannot have its
motion transformed into the B kind by being turned over
in any way. Suppose a body has the rotation x to y and
z to w. Turning it about the xy plane, we reverse the
direction of the motion x to y. But we also reverse the
z to w motion, for the point at the extremity of the
positive z axis is now at the extremity of the negative z
axis, and since we have not interfered with its motion it
goes in the direction of position w. Hence we have y to
x and iv to z, which is the same as x to y and z to w.
Thus both components are reversed, and there is the A
motion over again. The B kind is the semi-negative,
with only one component reversed.

Hence a system of molecules with the A motion would
not destroy it in one another, and would impart it to a
body in contact with them. Thus A and B motions
possess the first requisite which must be demanded in
any mode of motion representative of electricity.

Let us trace out the consequences of defining positive
electricity as an A motion and negative electricity as a B
niction. The combination of positive and negative

228 THE FOURTH DIMENSION

electricity produces a current. Imagine a vortex in the
ether of the A kind and unite with this one of the B kind.
An A motion and B motion produce rotation round a plane,
which is in the ether a vortex round an axial surface.
It is a vortex of the kind we rej)resent as a part of a
sphere turning inside out. Now such a vortex must have
its rim on a boundary of the ether — on a body in the
ether.

Let us suppose that a conductor is a body which has
the property of serving as the terminal abutment of such
a vortex. Then the conception we must form of a closed
current is of a vortex sheet having its edge along the
circuit of the conducting wire. The whole wire will then
be like the centres on which a spindle turns in three-
dimensional space, and any interruption of the continuity
of the wire will produce a tension in place of a continuous
revolution.

As the direction of the rotation of the vortex is from a
three-space direction into the fourth dimension and back
again, there will be no direction of flow to the current ;
but it will have two sides, according to whether z goes
to w or z goes to negative w.

We can draw any line from one part of the circuit to
another ; then the ether along that line is rotating round
its points.

This geometric image corresponds to the definition of
an electric circuit. It is known that the action does not
lie in the wire, but in the medium, arid it is known that
there is no direction of flow in the wire.

No explanation has been offered in three-dimensional
mechanics of how an action can be impressed throughout
a region and yet necessarily run itself out along a closed
boundary, as is the case in an electric current. But this
phenomenon corresponds exactly to the definition of a
four-dimensional vortex.

RECAPITULATION AND EXTENSION 229

If we take a very long magnet, so long that one of its
poles is practically isolated, and put this pole in the
vicinity of an electric circuit, we find that it moves.

Now, assuming for the sake of simplicity that the wire
which determines the current is in the form of a circle,
if we take a number of small magnets and place them all
pointing in the same direction normal to the plane of the
circle, so that they fill it and the wire binds them round,
we find that this sheet of magnets has the same effect on
the magnetic pole that the current has. The sheet of
magnets may be curved, but the edge of it must coincide
with the wire. The collection of magnets is then
equivalent to the vortex sheet, and an elementary magnet
to a part of it. Thus, we must think of a magnet as
conditioning a rotation in the ether round the plane
which bisects at right angles the line joining its poles.

If a current is started in a circuit, we must imagine
vortices like bowls turning themselves inside out, starting
from the contour. In reaching a parallel circuit, if the
vortex sheet were interrupted and joined momentarily to
the second circuit by a free rim, the axis plane would lie
between the two circuits, and a point on the second circuit
opposite a point on the first would correspond to a point
opposite to it on the first ; hence we should expect a
current in the opposite direction in the second circuit.
Thus the phenomena of induction are not inconsistent
with the hypothesis of a vortex about an axial plane.

In four-dimensional space, in which all four dimensions
were commensurable, the intensity of the action transmitted
by the medium would vary inversely as the cube of the
distance. Now, the action of a current on a magnetic
pole varies inversely as the square of the distance; hence,
over measurable distances the extension of the ether in
the fourth dimension cannot be assumed as other than
-mall in comparison with those distances.

230 THE FOURTH DIMENSION

If we suppose the ether to be filled with vortices in the
shape of four-dimensional spheres rotating with the A
motion, the B motion would correspond to electricity in
the one-fluid theory. There would thus be a possibility
of electricity existing in two forms, statically, by itself,
and, combined with the universal motion, in the form of
a current.

To arrive at a definite conclusion it will be necessary to
investigate the resultant pressures which accompany the
collocation of solid vortices with surface ones.

To recapitulate :

The movements and mechanics of four-dimensional
space are definite and intelligible. A vortex with a
surface as its axis affords a geometric image of a closed
circuit, and there are rotations which by their polarity
afford a possible definition of statical electricity.

APPENDIX

THE MODELS

In Chapter XI. a description has been given which will
enable any one to make a set of models illustrative of the
tesseract and its properties. The set here supposed to be
employed consists of : —

1 . Three sets of twenty-seven cubes each.

2. Twenty- seven slabs.

3. Twelve cubes with points, lines, faces, distinguished

by colours, which will be called the catalogue cubes.

The preparation of the twelve catalogue cubes involves
the expenditure of a considerable amount of time. It is
advantageous to use them, but they can be replaced by
the drawing of the views of the tesseract or by a reference
to figs. 103, 104, 105, 100 of the text.

The slabs are coloured like the twenty-seven cubes of
the first cubic block in fig. 101, the one with red,
white, yellow axes.

The colours of the three sets of twenty-seven cubes are
those of the cubes shown in fig. 101.

The slabs are used to form the representation of a cube
in a plane, and can well be dispensed with by any one
who is accustomed to deal with solid figures. But the
whole theory depends on a careful observation of how the
cube would be represented by these Blabs.

In the first step, that of forming a clear idea how a

231

232 THE FOURTH DIMENSION

plane being would represent three-dimensional Bpace, only
one of the catalogue cubes and one of the three block- lb
needed.

Application to the Step fkom Plane to Solid.

Look at fig. ] of the views of the tesseract, or, what
comes to the same thing, take catalogue cube No. 1 and
place it before you with the red line running up, the
white line running to the right, the yellow line running
away. The three dimensions of space are then marked
out by these lines or axes. Now take a piece of card-
board, or a book, and place it so that it forms a wall
extending up and down not opposite to you, but run-
ning away parallel to the wall of the room on your
left hand.

Placing the catalogue cube against this wall we see
that it comes into contact with it by the red and yellow
lines, and by the included orange face.

In the plane being's world the aspect he has of the
cube would be a square surrounded by red and yellow
lines with grey points.

Now, keeping the red line fixed, turn the cube about it
so that the yellow line goes out to the right, and the
white line comes into contact with the plane.

In this case a different aspect is presented to the plane
being, a square, namely, surrounded by red and white
lines and grey points. You should particularly notice
that when the yellow line goes out, at right angles to the
plane, and the white comes in, the latter does not run in
the same sense that the yellow did.

From the fixed grey point at the base of the red line
the yellow line ran away from you. The white line now
runs towards you. This turning at right angles makes
the line which was out of the plane before, come into it

ArPENDix 233

in an opposite sense to that in which the line ran which
has just left the plane. If the cube does not break
through the plane this is always the rule.

Again turn the cube back to the normal position with
red running up, white to the right, and yellow away, and
try another turning.

You can keep the yellow line fixed, and turn the cube
about it. In this case the red line going out to the
right the white line will come in pointing downwards.

You will be obliged to elevate the cube from the table
in order to carry out this turning. It is always necessary
when a vertical axis goes out of a space to imagine a
movable support which will allow the line which ran out
before to come in below.

Having looked at the three ways of turning the cube
so as to present different faces to the plane, examine what
would be the appearance if a square hole were cut in the
piece of cardboard, and the cube were to pass through it.
A hole can be actually cut, and it will be seen that in the
normal position, with red axis running up, yellow away,
and white to the right, the square first perceived by the
plane being — the one contained by red and yellow lines —
would be replaced by another square of which the line
towards you is pink — the section line of the pink face.
The line above is light yellow, below is light yellow and
on the opposite side away from you is pink.

In the same way the cube can be pushed through a
Bquare opening in the plane from any of the positions
which you have already turned it into. In each case
the plane being will perceive a different set of contour
line-.

Having observed these facts about the catalogue cube,
turn now to the first block of twenty-seven cubes.

You notice that the colour scheme on the catalogue cube
and that of this set of blocks is the same.

234 THE FOURTH DIMENSION

Place them before you, a grey or null cube on tin-
table, above it a red cube, and on the top a null cube
again. Then away from you place a yellow cube, and
beyond it a null cube. Then to the right place a white
cube and beyond it another null. Then complete the
block, according to the scheme of the catalogue cube,
putting in the centre of all an ochre cube.

You have now a cube like that which is described in
the text. For the sake of simplicity, in some cases, this
cubic block can be reduced to one of eight cubes, by
leaving out the terminations in each direction. Thus,
instead of null, red, null, three cubes, you can take null,
red. two cubes, and so on.

It is useful, however, to practise the representation in
a plane of a block of twenty-seven cubes. For this
purpose take the slabs, and build them up against the
piece of cardboard, or the book in such a way as to
represent the different aspects of the cube.

Proceed as follows : —

First, cube in normal position.

Place nine slabs against the cardboard to represent the
nine cubes in the wall of the red and yellow axes, facing
the cardboard ; these represent the aspect of the cube as it
touches the plane.

Now push these along the cardboard and make a
different set of nine slabs to represent the appearance
which the cube would present to a plane being, if it were
to pass half way through the plane.

There would be a white slab, above it a pink one, above
that another white one, and six others, representing what
would be the nature of a section across the middle of the
block of cubes. The section can be thought of as a thin
slice cut out by two parallel cuts across the cube.
Having arranged these nine slabs, push them along the
plane, and make another set of nine to represent what

APPENDIX 235

would be the appearance of the cube when it had almost
completely gone through. This set of nine will be the
same as the first set of nine.

Now we have in the plane three sets of nine slabs
each, which represent three sections of the twenty-seven
block.

They are put alongside one another. We see that it
does not matter in what order the sets of nine are put.
As the cube passes through the plane they represent ap-
pearances which follow the one after the other. If they
were what they represented, they could not exist in the
same plane together.

This is a rather important point, namely, to notice that
they should not co-exist on the plane, and that the order
in which they are placed is indifferent. When we
represent a four-dimensional body our solid cubes are to
us in the same position that the slabs are to the plane
being. You should also notice that each of these slabs
represents only the very thinnest slice of a cube. The
set of nine slabs first set up represents the side surface of
the block. It is, as it were, a kind of tray — a beginning
from which the solid cube goes off. The slabs as we use
them have thickness, but this thickness is a necessity of
construction. They are to be thought of as merely of the
thickness of a line.

If now the block of cubes passed through the plane at
the rate of an inch a minute the appearance to a plane
being would be represented by : —

1. The first set of nine slabs lasting for one minute.

2. The second set of nine slabs lasting for one minute.

3. The third set of nine slabs lasting for one minute.
Now the appearances which the cube would present

to the plane being in other positions can be shown by
means of these slabs. The use of such Blabs would be
the means by which a plane being could acquhv a

230 THE FOURTH DIMENSION

familiarity with our cube. Turn the catalogue cube (or
imagine the coloured figure turned) so that the red line
runs up, the yellow line out to the right, and the white
line towards you. Then turn the block of cubes to
occupy a similar position.

The block has now a different wall in contact with
the plane. Its appearance to a plane being will not be
the same as before. He has, however, enough slabs to
represent this new set of appearances. But he must
remodel his former arrangement of them.

He must take a null, a red, and a null slab from the first
of his sets of slabs, then a white, a pink, and a white from
the second, and then a null, a red, and a null from the
third set of slabs.

He takes the first column from the first set, the first
column from the second set, and the first column from
the third set.

To represent the half-way-through appearance, which
is as if a very thin slice were cut out half way through the
block, he must take the second column of each of his
sets of slabs, and to represent the final appearance, the
third column of each set.

Now turn the catalogue cube back to the normal
position, and also the block of cubes.

There is another turning — a turning about the yellow
line, in which the white axis comes below the support.

You cannot break through the surface of the table, so
you must imagine the old support to be raised. Then
the top of the block of cubes in its new position is at the
level at which the base of it was before.

Now representing the appearance on the plane, we must
draw a horizontal line to represent the old base. The
line should be drawn three inches high on the cardboard.

Below this the representative slabs can be arranged.

It is easy to see what they are. The old arrangements

APPENDIX 237

have to be broken up, and the layers taken in order, the
first layer of each for the representation of the aspect of
the block as it touches the plane.

Then the second layers will represent the appearance
half way through, and the third layers will represent the
final appearance.

It is evident that the slabs individually do not represent
the same portion of the cube in these different presenta-
tions.

In the first case each slab represents a section or a face
perpendicular to the white axis, in the second case a
face or a section which runs perpendicularly to the yellow
axis, and in the third case a section or a face perpendicular
to the red axis.

But by means of these nine slabs the plane being can
represent the whole of the cubic block. He can touch
and handle each portion of the cubic block, there is no
part of it which he cannot observe. Taking it bit by bit,
two axes at a time, he can examine the whole of it.

Our Kepresentation of a Block of Tesseracts.

Look at the views of the tesseract 1, 2, 3, or take the
catalogue cubes 1, 2, 3, and place them in front of you,
in any order, say running from left to right, placing 1 in
the normal position, the red axis running up, the white
to the right, and yellow away.

Now notice that in catalogue cube 2 the colours of each
region are derived from those of the corresponding region
of cube 1 by the addition of blue. Thus null -f- blue =
blue, and the corners of number 2 are blue. Again,
red + blue = purple, and the vertical lines of 2 are purple.
Blue + yellow = green, and the line which runs away is
coloured green.

By means of these observations you may be sure thai

238 THE FOURTH DIMENSION

catalogue cube 2 is rightly placed. Catalogue cube 3 is
just like number 1.

Having these cubes in what we may call their normal
position, proceed to build up the three sets of blocks.

This is easily done in accordance with the colour scheme
on the catalogue cubes.

The first block we already know. Build up the second
block, beginning with a blue corner cube, placing a purple
on it, and so on.

Having these three blocks we have the means of
representing the appearances of a group of eighty-one
tesseracts.

Let us consider a moment what the analogy in the case
of the plane being is.

He has his three sets of nine slabs each. We have our
three sets of twenty-seven cubes each.

Our cubes are like his slabs. As his slabs are not the
things which they represent to him, so our cubes are not
the things they represent to us.

The plane being's slabs are to him the faces of cubes.
Our cubes then are the faces of tesseracts, the cubes by
which they are in contact with our space.

As each set of slabs in the case of the plane being
might be considered as a sort of tray from which the solid
contents of the cubes came out, so our three blocks of
cubes may be considered as three-space trays, each of
which is the beginning of an inch of the solid contents
of the four-dimensional solids starting from them.

We want now to use the names null, red, white, etc.,
for tesseracts. The cubes we use are only tesseract faces.
Let us denote that fact by calling the cube of null colour,
null face ; or, shortly, null f., meaning that it is the face
of a tesseract.

To determine which face it is let us look at the catalogue
cube 1 or the first of the views of the tesseract, which

APPENDIX 239

can be used instead of the models. It has three axes,
red, white, yellow, in our space. Hence the cube deter-
mined by these axes is the face of the tesseract which we
now have before us. It is the ochre face. It is enough,
however, simply to say null f., red f. for the cubes which
we use.

To impress this in your mind, imagine that tesseracts
do actually run from each cube. Then, when you move the
cubes about, you move the tesseracts about with them.
You move the face but the tesseract follows with it, as the
cube follows when its face is shifted in a plane.

The cube null in the normal position is the cube which
has in it the red, yellow, white axes. It is the face
having these, but wanting the blue. In this way you can
define which face it is you are handling. I will write an
" f." after the name of each tesseract just as the plane
being might call each of his slabs null slab, yellow slab,
etc., to denote that they were representations.

We have then in the first block of twenty-seven cubes,
the following — null f., red f., null f., going up ; white f., null
f., lying to the right, aud so ou. Starting from the null
point and travelling up one inch we are in the null region,
the same for the away and the right-hand directions.
And if we were to travel in the fourth dimension for an
inch we should still be in a null region. The tesseract
stretches equally all four ways. Hence the appearance we
have in this first block would do equally well if the
tesseract block were to move across our space for a certain
distance. For anything less than an inch of their trans-
verse motion we should still have the same appearance.
You must notice, however, that we should not have null
face after the motion had begun.

When the tesseract, null for instance, had moved ever
so little we should not have a face of null but a section of
null in our space. Hence, when we think of the motion

240 THE FOURTH DIMENSION

across our space we must call our cubes tesseract sections.
Thus on null passing across we should see first null f., then
null s., and then, finally, null f. again.

Imagine now the whole first block of twenty-seven
tesseracts to have moved tranverse to our space a distance
of one inch. Then the second set of tesseracts, which
originally were an inch distant from our space, would be
ready to come in.

Their colours are shown in the second block of twenty-
seven cubes which you have before you. These represent
the tesseract faces of the set of tesseracts that lay before
an inch away from our space. They are ready now to
come in, and we can observe their colours. In the place
which null f. occupied before we have blue f., in place of
red f. we have purple f., and so on. Each tesseract is
coloured like the one whose place it takes in this motion
with the addition of blue.

Now if the tesseract block goes on moving at the rate
of an inch a minute, this next set of tesseracts will occupy
a minute in passing across. We shall see, to take the null
one for instance, first of all null face, then null section,
then null face again.

At the end of the second minute the second set of
tesseracts has gone through, and the third set comes in.
This, as you see, is coloured just like the first. Altogether,
these three sets extend three inches in the fourth dimension,
making the tesseract block of equal magnitude in all
dimensions.

We have now before us a complete catalogue of all the
tesseracts in our group. We have seen them all, and we
shall refer to this arrangement of the blocks as the
" normal position." We have seen as much of each
tesseract at a time as could be done in a three-dimen-
sional space. Each part of each tesseract has been in
our space, and we could have touched it.

APPENDIX 241

The fourth dimension appeared to us as the duration
of the block.

If a bit of our matter were to be subjected to the same
motion it would be instantly removed out of our space.
Being thin in the fourth dimension it is at once taken
out of our space by a motion in the fourth dimension.

But the tesseract block we represent having length in
the fourth dimension remains steadily before our eyes for
three minutes, when it is subjected to this transverse
motion.

We have now to form representations of the other
views of the same tesseract group which are possible in
our space.

Let us then turn the block of tesseracts so that another
face of it comes into contact with our space, and then
by observing what we have, and what changes come when
the block traverses our space, we shall have another view
of it. The dimension which appeared as duration before
will become extension in one of our known dimensions,
and a dimension which coincided with one of our space
dimensions will appear as duration.

Leaving catalogue cube 1 in the normal position,
remove the other two, or suppose them removed. We
have in space the red, the yellow, and the white axes.
Let the white axis go out into the unknown, and occupy
the position the blue axis holds. Then the blue axis,
which runs in that direction now will come into space.
But it will not come in pointing in the same way that
the white axis does now. It will point in the opposite
sense. It will come in running to the left instead of
running to the right as the white axis does now.

When tin's turning take- place every pari of the cube 1
will disappear except the left-hand face — the orange Face.

And the new cube that appears in our space will run to
the left from this orange face, having axes, red, yellow, blue.

242 THE FOURTH DIMENSION

Take models 4, 5, G. Place 4, or suppose No. 4 of the
tesseract views placed, with its orange face coincident with
the orange face of 1, red line to red line, and yellow line
to yellow line, with the blue line pointing to the left.
Then remove cube 1 and we have the tesseract face
which comes in when the white axis runs in the positive
unknown, and the blue axis comes into our space.

Now place catalogue cube 5 in some position, it does
not matter which, say to the left ; and place it so that
there is a correspondence of colour corresponding to the
colour of the line that runs out of space. The line that
runs out of space is white, hence, every part of this
cube 5 should differ from the corresponding part of 4 by
an alteration in the direction of white.

Thus we have white points in 5 corresponding to the
null points in 4. We have a pink line corresponding to
a red line, a light yellow line corresponding to a yellow
line, an ochre face corresponding to an orange face. This
cube section is completely named in Chapter XL Finally
cube 6 is a replica of 1.

These catalogue cubes will enable us to set up our
models of the block of tesseracts.

First of all for the set of tesseracts, which beginning
in our space reach out one inch in the unknown, we have
the pattern of catalogue cube 4.

We see that we can build up a block of twenty-seven
tesseract faces after the colour scheme of cube 4, by
taking the left-hand wall of block 1, then the left-hand
wall of block 2, and finally that of block 3. We take,
that is, the three first walls of our previous arrangement
to form the first cubic block of this new one.

This will represent the cubic faces by which the group
of tesseracts in its new position touches our space.
We have running up, null f., red f., null f. In the next
vertical line, on the side remote from us, we have yellow f.,

APPENDIX 243

orange f., yellow f., and then the first colours over again.
Then the three following columns are, blue f., purple f.,
blue f. ; green f., brown f., green f. ; blue f., purple f., blue f.
The last three columns are like the first.

These tesseract s touch our space, and none of them are
by any part of them distant more than an inch from it.
What lies beyond them in the unknown ?

This can be told by looking at catalogue cube 5.
According to its scheme of colour we see that the second
wall of each of our old arrangements must be taken.
Putting them together we have, as the corner, white f.
above it, pink f. above it, white f. The column next to
this remote from us is as follows : — light yellow f., ochre f.,
light yellow f., and beyond this a column like the first.
Then for the middle of the block, light blue f., above
it light purple, then light blue. The centre column has,
at the bottom, light green f., light brown f. in the centre
and at the top light green f. The last wall is like the
first.

The third block is made by taking the third walls of
our previous arrangement, which we called the normal
one.

You may ask what faces and what sections our cubes
represent. To answer this question look at what axes
you have in our space. You have red, yellow, blue.
Now these determine brown. The colours red,
yellow, blue are supposed by us when mixed to produce
a brown colour. And that cube which is determined
by the red, yellow, blue axes we call the brown cube.

When the tesseract block in its new position begins to
move across our space each tesseract in it gives a section
in our space. This section is transverse to the white
axis, which now runs in the unknown.

As the tesseract in it- present position passes across
our space, we should see first of all the first of the blocks

244 THE FOURTH DIMENSION

of cubic faces we have put up — these would last for a
minute, then would come the second block and then the
third. At first we should have a cube of tesseract faces,
each of which would be brown. Directly the movement
began, we should have tesseract sections transverse to the
white line.

There are two more analogous positions in which the
block of tesseracts can be placed. To find the third
position, restore the blocks to the normal arrangement.

Let us make the yellow axis go out into the positive
unknown, and let the blue axis, consequently, come in
running towards us. The yellow ran away, so the blue
will come in running towards us.

Put catalogue cube 1 in its normal position. Take
catalogue cube 7 and place it so that its pink face
coincides with the pink face of cube 1, making also its
red axis coincide with the red axis of 1 and its white
with the white. Moreover, make cube 7 come
towards us from cube 1. Looking at it we see in our
space, red, white, and blue axes. The yellow runs out.
Place catalogue cube 8 in the neighbourhood of
7 — observe that every region in 8 has a change in
the direction of yellow from the corresponding region
in 7. This is because it represents what you come
to now in going in the unknown, when the yellow axis
runs out of our space. Finally catalogue cube 9,
which is like number 7, shows the colours of the third
set of tesseracts. Now evidently, starting from the
normal position, to make up our three blocks of tesseract
faces we have to take the near wall from the first block,
the near wall from the second, and then the near wall
from the third block. This gives us the cubic block
formed by the faces of the twenty-seven tesseracts which
are now immediately touching our space.

Following the colour scheme of catalogue cube 8,

APPENDIX 245

we make the next set of twenty-seven tesseract faces,
representing the tesseracts, each of which begins one inch
off from our space, by putting the second walls of our
previous arrangement together, and the representation
of the third set of tessaracts is the cubic block formed of
the remaining three walls.

Since we have red, white, blue axes in our space to
begin with, the cubes we see at first are light purple
tesseract faces, and after the transverse motion begins
we have cubic sections transverse to the yellow line.

Restore the blocks to the normal position, there
remains the case in which the red axis turns out of
space. In this case the blue axis will come in down-
wards, opposite to the sense in which the red axis ran.

In this case take catalogue cubes 10, 11, 12. Lift up
catalogue cube 1 and put 10 underneath it, imagining
that it goes down from the previous position of 1.

We have to keep in space the white and the yellow
axes, and let the red go out, the blue come in.

Now, you will find on cube 10 a light yellow face; this
should coincide with the base of 1, and the white and
yellow lines on the two cubes should coincide. Then the
blue axis running down you have the catalogue cube
correctly placed, and it forms a guide for putting up the
first representative block.

Catalogue cube 1 1 will represent what lies in the fourth
dimension — now the red line runs in the fourth dimen-
sion. Thus the change from 10 to 11 should be towards
red, corresponding to a null point is a red point, to a
white line is a pink line, to a yellow line an orange
line, and so on.

Catalogue cube 12 is like 10. Hence we see that to
build up our blocks of tesseract faces we must take the
bottom layer of the first block, hold that up in the air,
underneath it place the bottom layer of the second block,

246 THE FOURTH DIMENSION

and finally underneath this last the bottom layer of tin*
last of our normal blocks.

Similarly we make the second representative group by
taking the middle courses of our three blocks. The last
is made by taking the three topmost layers. The three
axes in our space before the transverse motion begins are
blue, white, yellow, so we have light green tesseract
faces, and after the motion begins sections transverse to
the red light.

These three blocks represent the appearances as the
tesseract group in its new position passes across our space.
The cubes of contact in this case are those determinal by
the three axes in our space, namely, the white, the
yellow, the blue. Hence they are light green.

It follows from this that light green is the interior
<mbe of the first block of representative cubic faces.

Practice in the manipulations described, with a
realization in each case of the face or section which
is in our space, is one of the best means of a thorough
-comprehension of the subject.

We have to learn how to get any part of these four-
dimensional figures into space, so that we can look at
them. We must first learn to swing a tesseract, and a
group of tesseract s about in any way.

When these operations have been repeated and the
method of arrangement of the set of blocks has become
familiar, it is a good plan to rotate the axes of the normal
cube 1 about a diagonal, and then repeat the whole series
of turnings.

Thus, in the normal position, red goes up, white to the
right, yellow away. Make white go up, yellow to the right,
and red away. Learn the cube in this position by putting
up the set of blocks of the normal cube, over and over
again till it becomes as familiar to you as in the normal
position. Then when this is learned, and the corre-

APPENDIX 24?

sponding changes in the arrangements of the tesseract
groups are made, another change should he made : let,
in the normal cube, yellow go up, red to the right, and
white away.

Learn the normal block of cubes in this new position
by arranging them and re-arranging them till you know
without thought where each one goes. Then carry out
all the tesseract arrangements and turnings.

If you want to understand the subject, but do not see
your way clearly, if it does not seem natural and easy to
you, practise these turnings. Practise, first of all, the
turning of a block of cubes round, so that you know it
in every position as well as in the normal one. Practise
by gradually putting up the set of cubes in their new
arrangements. Then put up the tesseract blocks in their
arrangements. This will give you a working conception
of higher space, you will gain the feeling of it, whether
you take up the mathematical treatment of it or not.

• i by ffazell, Walton <( Viney, L''., London and Aylesbury.

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THE INSTITUTE OF HEPIAFVAl STUI

10 ELMSLEY FLACC

iOnw^lO 5, CANADA,

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